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Номер 78, страница 25 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
I. Уравнения, неравенства с двумя переменными и их системы. 1. Нелинейные уравнения с двумя переменными - номер 78, страница 25.
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=> null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. 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=> null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. 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#escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Branch {#1126 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_page_start" => "18" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1128 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157 …2} ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_page_start" => "18" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1128 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107640 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1. Нелинейные уравнения с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095} "field_page_start" => "19" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:3 [ 0 => App\Models\Element {#1159 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1276433 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Edition {#1161 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1107774" "type" => "task" ] "text" => "<p><strong>78.</strong> Постройте на координатной плоскости множество всех точек, координаты $(x; y)$ которых являются решениями уравнения:</p><p><strong>a)</strong> $(x-y)^2 = 2x + 2y - x^2 - y^2 - 2$; <strong>в)</strong> $2x^2 + 5xy - 12y^2 = 0$;</p><p><strong>б)</strong> $\frac{x^2 - y^2}{x - y} = \frac{x^2 + y^2}{x + y}$; <strong>г)</strong> $x^2 + y^2 = 6|y| + 16$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1554" "height" => 397 "path" => "/media/algebra_09/soltan-u19/0-1/78-1.webp?ts=1752834155" ] ] ] #original: array:7 [ "id" => 1276433 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160} "task" => array:2 [ "refs" => "1107774" "type" => "task" ] "text" => "<p><strong>78.</strong> Постройте на координатной плоскости множество всех точек, координаты $(x; y)$ которых являются решениями уравнения:</p><p><strong>a)</strong> $(x-y)^2 = 2x + 2y - x^2 - y^2 - 2$; <strong>в)</strong> $2x^2 + 5xy - 12y^2 = 0$;</p><p><strong>б)</strong> $\frac{x^2 - y^2}{x - y} = \frac{x^2 + y^2}{x + y}$; <strong>г)</strong> $x^2 + y^2 = 6|y| + 16$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1554" "height" => 397 "path" => "/media/algebra_09/soltan-u19/0-1/78-1.webp?ts=1752834155" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1167 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1277643 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1168 #items: array:1 [ 0 => App\Models\Edition {#1169 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1107774" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1413" "height" => 1988 "path" => "/media/algebra_09/soltan-u19/1-1/78-1.webp?ts=1752830799" ] 1 => array:5 [ "name" => "78-2.jpg" "alt" => null "width" => "1401" "height" => 1279 "path" => "/media/algebra_09/soltan-u19/1-1/78-2.webp?ts=1752830799" ] ] ] #original: array:6 [ "id" => 1277643 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1168} "task" => array:2 [ "refs" => "1107774" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "78-1.jpg" "alt" => null "width" => "1413" "height" => 1988 "path" => "/media/algebra_09/soltan-u19/1-1/78-1.webp?ts=1752830799" ] 1 => array:5 [ "name" => "78-2.jpg" "alt" => null "width" => "1401" "height" => 1279 "path" => "/media/algebra_09/soltan-u19/1-1/78-2.webp?ts=1752830799" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1175 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1553679 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:1 [ 0 => App\Models\Edition {#1177 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1107774" "type" => "task" ] "text" => "<p><strong>а)</strong> Исходное уравнение: $(x - y)^2 = 2x + 2y - x^2 - y^2 - 2$.</p><p>Раскроем скобки в левой части и перенесем все слагаемые из правой части в левую:<br>$x^2 - 2xy + y^2 = 2x + 2y - x^2 - y^2 - 2$<br>$x^2 - 2xy + y^2 - 2x - 2y + x^2 + y^2 + 2 = 0$</p><p>Приведем подобные слагаемые:<br>$2x^2 + 2y^2 - 2xy - 2x - 2y + 2 = 0$</p><p>Разделим обе части уравнения на 2:<br>$x^2 + y^2 - xy - x - y + 1 = 0$</p><p>Чтобы упростить это уравнение, умножим его снова на 2 и сгруппируем слагаемые так, чтобы выделить полные квадраты:<br>$2x^2 + 2y^2 - 2xy - 2x - 2y + 2 = 0$<br>$(x^2 - 2xy + y^2) + (x^2 - 2x + 1) + (y^2 - 2y + 1) = 0$</p><p>Теперь мы можем свернуть каждую группу в полный квадрат:<br>$(x - y)^2 + (x - 1)^2 + (y - 1)^2 = 0$</p><p>Сумма квадратов действительных чисел равна нулю тогда и только тогда, когда каждое из этих чисел равно нулю. Таким образом, мы получаем систему уравнений:<br>$x - y = 0$<br>$x - 1 = 0$<br>$y - 1 = 0$</p><p>Из второго и третьего уравнений находим, что $x=1$ и $y=1$. Эти значения удовлетворяют и первому уравнению: $1 - 1 = 0$.Следовательно, решением является единственная точка с координатами $(1; 1)$.</p><p><strong>Ответ:</strong> Множество точек является единственной точкой $(1; 1)$.</p><p><strong>б)</strong> Исходное уравнение: $\frac{x^2 - y^2}{x - y} = \frac{x^2 + y^2}{x + y}$.</p><p>Найдем область допустимых значений (ОДЗ). Знаменатели дробей не должны быть равны нулю:<br>$x - y \neq 0 \Rightarrow x \neq y$<br>$x + y \neq 0 \Rightarrow x \neq -y$</p><p>Это означает, что искомое множество точек не может лежать на прямых $y=x$ и $y=-x$.</p><p>Упростим левую часть уравнения, используя формулу разности квадратов $x^2 - y^2 = (x - y)(x + y)$:<br>$\frac{(x - y)(x + y)}{x - y} = x + y$</p><p>Теперь уравнение принимает вид:<br>$x + y = \frac{x^2 + y^2}{x + y}$</p><p>Умножим обе части на $(x + y)$, так как по ОДЗ $x+y \neq 0$:<br>$(x + y)^2 = x^2 + y^2$<br>$x^2 + 2xy + y^2 = x^2 + y^2$</p><p>Вычтем $x^2 + y^2$ из обеих частей:<br>$2xy = 0$</p><p>Это уравнение выполняется, если $x = 0$ или $y = 0$. Геометрически это соответствует двум координатным осям: оси ординат (прямая $x=0$) и оси абсцисс (прямая $y=0$).</p><p>Теперь учтем ОДЗ. Точки, в которых $x=y$ или $x=-y$, должны быть исключены.<br>На прямой $x=0$ условие $x \neq y$ означает $0 \neq y$, а условие $x \neq -y$ означает $0 \neq -y$. Оба условия означают, что $y \neq 0$.<br>На прямой $y=0$ условие $x \neq y$ означает $x \neq 0$, а условие $x \neq -y$ означает $x \neq 0$.<br>В обоих случаях точка $(0; 0)$ должна быть исключена.</p><p><strong>Ответ:</strong> Множество точек — это объединение координатных осей $x=0$ и $y=0$ за исключением точки их пересечения — начала координат $(0; 0)$.</p><p><strong>в)</strong> Исходное уравнение: $2x^2 + 5xy - 12y^2 = 0$.</p><p>Это однородное уравнение второй степени. Разложим левую часть на множители. Для этого представим средний член $5xy$ в виде суммы $8xy - 3xy$:<br>$2x^2 + 8xy - 3xy - 12y^2 = 0$</p><p>Сгруппируем слагаемые и вынесем общие множители за скобки:<br>$(2x^2 + 8xy) - (3xy + 12y^2) = 0$<br>$2x(x + 4y) - 3y(x + 4y) = 0$</p><p>Теперь вынесем за скобки общий множитель $(x + 4y)$:<br>$(2x - 3y)(x + 4y) = 0$</p><p>Произведение равно нулю, если хотя бы один из множителей равен нулю. Таким образом, уравнение распадается на два:<br>1) $2x - 3y = 0 \Rightarrow 3y = 2x \Rightarrow y = \frac{2}{3}x$<br>2) $x + 4y = 0 \Rightarrow x = -4y \Rightarrow y = -\frac{1}{4}x$</p><p>Каждое из этих уравнений задает прямую, проходящую через начало координат.</p><p><strong>Ответ:</strong> Множество точек — это пара прямых $y = \frac{2}{3}x$ и $y = -\frac{1}{4}x$, пересекающихся в начале координат.</p><p><strong>г)</strong> Исходное уравнение: $x^2 + y^2 = 6|y| + 16$.</p><p>Наличие модуля $|y|$ требует рассмотрения двух случаев.</p><p><strong>Случай 1: $y \ge 0$</strong><br>В этом случае $|y| = y$, и уравнение принимает вид:<br>$x^2 + y^2 = 6y + 16$<br>Перенесем слагаемые с $y$ в левую часть и выделим полный квадрат:<br>$x^2 + y^2 - 6y = 16$<br>$x^2 + (y^2 - 6y + 9) = 16 + 9$<br>$x^2 + (y - 3)^2 = 25$<br>$x^2 + (y - 3)^2 = 5^2$<br>Это уравнение окружности с центром в точке $(0; 3)$ и радиусом $R=5$. Мы должны взять только ту часть этой окружности, которая удовлетворяет условию $y \ge 0$. Эта часть представляет собой дугу, расположенную над осью абсцисс и на самой оси.</p><p><strong>Случай 2: $y < 0$</strong><br>В этом случае $|y| = -y$, и уравнение принимает вид:<br>$x^2 + y^2 = 6(-y) + 16$<br>$x^2 + y^2 = -6y + 16$<br>Перенесем слагаемые с $y$ в левую часть и выделим полный квадрат:<br>$x^2 + y^2 + 6y = 16$<br>$x^2 + (y^2 + 6y + 9) = 16 + 9$<br>$x^2 + (y + 3)^2 = 25$<br>$x^2 + (y + 3)^2 = 5^2$<br>Это уравнение окружности с центром в точке $(0; -3)$ и радиусом $R=5$. Мы должны взять только ту часть этой окружности, которая удовлетворяет условию $y < 0$. Эта часть представляет собой дугу, расположенную под осью абсцисс.</p><p>Объединяя оба случая, получаем фигуру, состоящую из двух дуг окружностей. Обе окружности проходят через точки, где $y=0$:<br>$x^2 + (0 - 3)^2 = 25 \Rightarrow x^2 + 9 = 25 \Rightarrow x^2 = 16 \Rightarrow x = \pm 4$.<br>Таким образом, обе дуги соединяются в точках $(-4; 0)$ и $(4; 0)$.</p><p><strong>Ответ:</strong> Множество точек является объединением двух дуг окружностей:<br>1. Дуги окружности $x^2 + (y - 3)^2 = 25$ при $y \ge 0$.<br>2. Дуги окружности $x^2 + (y + 3)^2 = 25$ при $y < 0$.<br>Эти дуги образуют замкнутую кривую, симметричную относительно обеих координатных осей.</p>" ] #original: array:6 [ "id" => 1553679 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1176} "task" => array:2 [ "refs" => "1107774" "type" => "task" ] "text" => "<p><strong>а)</strong> Исходное уравнение: $(x - y)^2 = 2x + 2y - x^2 - y^2 - 2$.</p><p>Раскроем скобки в левой части и перенесем все слагаемые из правой части в левую:<br>$x^2 - 2xy + y^2 = 2x + 2y - x^2 - y^2 - 2$<br>$x^2 - 2xy + y^2 - 2x - 2y + x^2 + y^2 + 2 = 0$</p><p>Приведем подобные слагаемые:<br>$2x^2 + 2y^2 - 2xy - 2x - 2y + 2 = 0$</p><p>Разделим обе части уравнения на 2:<br>$x^2 + y^2 - xy - x - y + 1 = 0$</p><p>Чтобы упростить это уравнение, умножим его снова на 2 и сгруппируем слагаемые так, чтобы выделить полные квадраты:<br>$2x^2 + 2y^2 - 2xy - 2x - 2y + 2 = 0$<br>$(x^2 - 2xy + y^2) + (x^2 - 2x + 1) + (y^2 - 2y + 1) = 0$</p><p>Теперь мы можем свернуть каждую группу в полный квадрат:<br>$(x - y)^2 + (x - 1)^2 + (y - 1)^2 = 0$</p><p>Сумма квадратов действительных чисел равна нулю тогда и только тогда, когда каждое из этих чисел равно нулю. Таким образом, мы получаем систему уравнений:<br>$x - y = 0$<br>$x - 1 = 0$<br>$y - 1 = 0$</p><p>Из второго и третьего уравнений находим, что $x=1$ и $y=1$. Эти значения удовлетворяют и первому уравнению: $1 - 1 = 0$.Следовательно, решением является единственная точка с координатами $(1; 1)$.</p><p><strong>Ответ:</strong> Множество точек является единственной точкой $(1; 1)$.</p><p><strong>б)</strong> Исходное уравнение: $\frac{x^2 - y^2}{x - y} = \frac{x^2 + y^2}{x + y}$.</p><p>Найдем область допустимых значений (ОДЗ). Знаменатели дробей не должны быть равны нулю:<br>$x - y \neq 0 \Rightarrow x \neq y$<br>$x + y \neq 0 \Rightarrow x \neq -y$</p><p>Это означает, что искомое множество точек не может лежать на прямых $y=x$ и $y=-x$.</p><p>Упростим левую часть уравнения, используя формулу разности квадратов $x^2 - y^2 = (x - y)(x + y)$:<br>$\frac{(x - y)(x + y)}{x - y} = x + y$</p><p>Теперь уравнение принимает вид:<br>$x + y = \frac{x^2 + y^2}{x + y}$</p><p>Умножим обе части на $(x + y)$, так как по ОДЗ $x+y \neq 0$:<br>$(x + y)^2 = x^2 + y^2$<br>$x^2 + 2xy + y^2 = x^2 + y^2$</p><p>Вычтем $x^2 + y^2$ из обеих частей:<br>$2xy = 0$</p><p>Это уравнение выполняется, если $x = 0$ или $y = 0$. Геометрически это соответствует двум координатным осям: оси ординат (прямая $x=0$) и оси абсцисс (прямая $y=0$).</p><p>Теперь учтем ОДЗ. Точки, в которых $x=y$ или $x=-y$, должны быть исключены.<br>На прямой $x=0$ условие $x \neq y$ означает $0 \neq y$, а условие $x \neq -y$ означает $0 \neq -y$. Оба условия означают, что $y \neq 0$.<br>На прямой $y=0$ условие $x \neq y$ означает $x \neq 0$, а условие $x \neq -y$ означает $x \neq 0$.<br>В обоих случаях точка $(0; 0)$ должна быть исключена.</p><p><strong>Ответ:</strong> Множество точек — это объединение координатных осей $x=0$ и $y=0$ за исключением точки их пересечения — начала координат $(0; 0)$.</p><p><strong>в)</strong> Исходное уравнение: $2x^2 + 5xy - 12y^2 = 0$.</p><p>Это однородное уравнение второй степени. Разложим левую часть на множители. Для этого представим средний член $5xy$ в виде суммы $8xy - 3xy$:<br>$2x^2 + 8xy - 3xy - 12y^2 = 0$</p><p>Сгруппируем слагаемые и вынесем общие множители за скобки:<br>$(2x^2 + 8xy) - (3xy + 12y^2) = 0$<br>$2x(x + 4y) - 3y(x + 4y) = 0$</p><p>Теперь вынесем за скобки общий множитель $(x + 4y)$:<br>$(2x - 3y)(x + 4y) = 0$</p><p>Произведение равно нулю, если хотя бы один из множителей равен нулю. Таким образом, уравнение распадается на два:<br>1) $2x - 3y = 0 \Rightarrow 3y = 2x \Rightarrow y = \frac{2}{3}x$<br>2) $x + 4y = 0 \Rightarrow x = -4y \Rightarrow y = -\frac{1}{4}x$</p><p>Каждое из этих уравнений задает прямую, проходящую через начало координат.</p><p><strong>Ответ:</strong> Множество точек — это пара прямых $y = \frac{2}{3}x$ и $y = -\frac{1}{4}x$, пересекающихся в начале координат.</p><p><strong>г)</strong> Исходное уравнение: $x^2 + y^2 = 6|y| + 16$.</p><p>Наличие модуля $|y|$ требует рассмотрения двух случаев.</p><p><strong>Случай 1: $y \ge 0$</strong><br>В этом случае $|y| = y$, и уравнение принимает вид:<br>$x^2 + y^2 = 6y + 16$<br>Перенесем слагаемые с $y$ в левую часть и выделим полный квадрат:<br>$x^2 + y^2 - 6y = 16$<br>$x^2 + (y^2 - 6y + 9) = 16 + 9$<br>$x^2 + (y - 3)^2 = 25$<br>$x^2 + (y - 3)^2 = 5^2$<br>Это уравнение окружности с центром в точке $(0; 3)$ и радиусом $R=5$. Мы должны взять только ту часть этой окружности, которая удовлетворяет условию $y \ge 0$. Эта часть представляет собой дугу, расположенную над осью абсцисс и на самой оси.</p><p><strong>Случай 2: $y < 0$</strong><br>В этом случае $|y| = -y$, и уравнение принимает вид:<br>$x^2 + y^2 = 6(-y) + 16$<br>$x^2 + y^2 = -6y + 16$<br>Перенесем слагаемые с $y$ в левую часть и выделим полный квадрат:<br>$x^2 + y^2 + 6y = 16$<br>$x^2 + (y^2 + 6y + 9) = 16 + 9$<br>$x^2 + (y + 3)^2 = 25$<br>$x^2 + (y + 3)^2 = 5^2$<br>Это уравнение окружности с центром в точке $(0; -3)$ и радиусом $R=5$. Мы должны взять только ту часть этой окружности, которая удовлетворяет условию $y < 0$. Эта часть представляет собой дугу, расположенную под осью абсцисс.</p><p>Объединяя оба случая, получаем фигуру, состоящую из двух дуг окружностей. Обе окружности проходят через точки, где $y=0$:<br>$x^2 + (0 - 3)^2 = 25 \Rightarrow x^2 + 9 = 25 \Rightarrow x^2 = 16 \Rightarrow x = \pm 4$.<br>Таким образом, обе дуги соединяются в точках $(-4; 0)$ и $(4; 0)$.</p><p><strong>Ответ:</strong> Множество точек является объединением двух дуг окружностей:<br>1. Дуги окружности $x^2 + (y - 3)^2 = 25$ при $y \ge 0$.<br>2. Дуги окружности $x^2 + (y + 3)^2 = 25$ при $y < 0$.<br>Эти дуги образуют замкнутую кривую, симметричную относительно обеих координатных осей.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1107775" "type" => "task" ] "previous" => array:2 [ "refs" => "1107773" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Book {#1184 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1185 #items: array:1 [ 0 => App\Models\Term {#1186 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Term {#1188 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Term {#1190 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:3 [ 0 => App\Models\Term {#1192 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1193 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1194 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1195 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Term {#1197 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Term {#1199 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\Term {#1201 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1202 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1203 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1204 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1205 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1206 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1207 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1208 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1209 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1210 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1211 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1185} "field_class" => Illuminate\Database\Eloquent\Collection {#1187} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1189} "field_author" => Illuminate\Database\Eloquent\Collection {#1191} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1195} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1196} "field_country" => Illuminate\Database\Eloquent\Collection {#1198} "field_city" => Illuminate\Database\Eloquent\Collection {#1200} "field_series" => Illuminate\Database\Eloquent\Collection {#1202} "field_umk" => Illuminate\Database\Eloquent\Collection {#1203} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1204} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1205} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1206} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1207} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1208} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1209} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1210} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1211} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1418 #items: array:1 [ 0 => App\Models\BookPage {#1215 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1097767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-25" "field_display_title" => "25" "field_folder" => "folder1" "field_image_name" => "25" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1219 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ 0 => App\Models\Book {#1220 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1223 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1225 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1231 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1232 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1234 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1236 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1238 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1239 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1240 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1241 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1244 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1245 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1246 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1223 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1225 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1231 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1232 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1234 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1236 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1238 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1239 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1240 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1241 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1244 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1245 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1246 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1248 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1249 #items: array:1 [ 0 => App\Models\Element {#1250 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1260918 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1251 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260918 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1251 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097768" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097766" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1258 #items: array:8 [ 0 => App\Models\Task {#1259 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-71" "field_display_title" => "71" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1260 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1262 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1263 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1268 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1324 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "25" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-71" "field_display_title" => "71" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1260 …2} "field_metatags_title" => null "field_metatags_description" => null 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№78 (с. 25)
Условие. №78 (с. 25)
скриншот условия
78. Постройте на координатной плоскости множество всех точек, координаты $(x; y)$ которых являются решениями уравнения:
a) $(x-y)^2 = 2x + 2y - x^2 - y^2 - 2$; в) $2x^2 + 5xy - 12y^2 = 0$;
б) $\frac{x^2 - y^2}{x - y} = \frac{x^2 + y^2}{x + y}$; г) $x^2 + y^2 = 6|y| + 16$.
Решение. №78 (с. 25)
Решение 2 (rus). №78 (с. 25)
а) Исходное уравнение: $(x - y)^2 = 2x + 2y - x^2 - y^2 - 2$.
Раскроем скобки в левой части и перенесем все слагаемые из правой части в левую:
$x^2 - 2xy + y^2 = 2x + 2y - x^2 - y^2 - 2$
$x^2 - 2xy + y^2 - 2x - 2y + x^2 + y^2 + 2 = 0$
Приведем подобные слагаемые:
$2x^2 + 2y^2 - 2xy - 2x - 2y + 2 = 0$
Разделим обе части уравнения на 2:
$x^2 + y^2 - xy - x - y + 1 = 0$
Чтобы упростить это уравнение, умножим его снова на 2 и сгруппируем слагаемые так, чтобы выделить полные квадраты:
$2x^2 + 2y^2 - 2xy - 2x - 2y + 2 = 0$
$(x^2 - 2xy + y^2) + (x^2 - 2x + 1) + (y^2 - 2y + 1) = 0$
Теперь мы можем свернуть каждую группу в полный квадрат:
$(x - y)^2 + (x - 1)^2 + (y - 1)^2 = 0$
Сумма квадратов действительных чисел равна нулю тогда и только тогда, когда каждое из этих чисел равно нулю. Таким образом, мы получаем систему уравнений:
$x - y = 0$
$x - 1 = 0$
$y - 1 = 0$
Из второго и третьего уравнений находим, что $x=1$ и $y=1$. Эти значения удовлетворяют и первому уравнению: $1 - 1 = 0$.Следовательно, решением является единственная точка с координатами $(1; 1)$.
Ответ: Множество точек является единственной точкой $(1; 1)$.
б) Исходное уравнение: $\frac{x^2 - y^2}{x - y} = \frac{x^2 + y^2}{x + y}$.
Найдем область допустимых значений (ОДЗ). Знаменатели дробей не должны быть равны нулю:
$x - y \neq 0 \Rightarrow x \neq y$
$x + y \neq 0 \Rightarrow x \neq -y$
Это означает, что искомое множество точек не может лежать на прямых $y=x$ и $y=-x$.
Упростим левую часть уравнения, используя формулу разности квадратов $x^2 - y^2 = (x - y)(x + y)$:
$\frac{(x - y)(x + y)}{x - y} = x + y$
Теперь уравнение принимает вид:
$x + y = \frac{x^2 + y^2}{x + y}$
Умножим обе части на $(x + y)$, так как по ОДЗ $x+y \neq 0$:
$(x + y)^2 = x^2 + y^2$
$x^2 + 2xy + y^2 = x^2 + y^2$
Вычтем $x^2 + y^2$ из обеих частей:
$2xy = 0$
Это уравнение выполняется, если $x = 0$ или $y = 0$. Геометрически это соответствует двум координатным осям: оси ординат (прямая $x=0$) и оси абсцисс (прямая $y=0$).
Теперь учтем ОДЗ. Точки, в которых $x=y$ или $x=-y$, должны быть исключены.
На прямой $x=0$ условие $x \neq y$ означает $0 \neq y$, а условие $x \neq -y$ означает $0 \neq -y$. Оба условия означают, что $y \neq 0$.
На прямой $y=0$ условие $x \neq y$ означает $x \neq 0$, а условие $x \neq -y$ означает $x \neq 0$.
В обоих случаях точка $(0; 0)$ должна быть исключена.
Ответ: Множество точек — это объединение координатных осей $x=0$ и $y=0$ за исключением точки их пересечения — начала координат $(0; 0)$.
в) Исходное уравнение: $2x^2 + 5xy - 12y^2 = 0$.
Это однородное уравнение второй степени. Разложим левую часть на множители. Для этого представим средний член $5xy$ в виде суммы $8xy - 3xy$:
$2x^2 + 8xy - 3xy - 12y^2 = 0$
Сгруппируем слагаемые и вынесем общие множители за скобки:
$(2x^2 + 8xy) - (3xy + 12y^2) = 0$
$2x(x + 4y) - 3y(x + 4y) = 0$
Теперь вынесем за скобки общий множитель $(x + 4y)$:
$(2x - 3y)(x + 4y) = 0$
Произведение равно нулю, если хотя бы один из множителей равен нулю. Таким образом, уравнение распадается на два:
1) $2x - 3y = 0 \Rightarrow 3y = 2x \Rightarrow y = \frac{2}{3}x$
2) $x + 4y = 0 \Rightarrow x = -4y \Rightarrow y = -\frac{1}{4}x$
Каждое из этих уравнений задает прямую, проходящую через начало координат.
Ответ: Множество точек — это пара прямых $y = \frac{2}{3}x$ и $y = -\frac{1}{4}x$, пересекающихся в начале координат.
г) Исходное уравнение: $x^2 + y^2 = 6|y| + 16$.
Наличие модуля $|y|$ требует рассмотрения двух случаев.
Случай 1: $y \ge 0$
В этом случае $|y| = y$, и уравнение принимает вид:
$x^2 + y^2 = 6y + 16$
Перенесем слагаемые с $y$ в левую часть и выделим полный квадрат:
$x^2 + y^2 - 6y = 16$
$x^2 + (y^2 - 6y + 9) = 16 + 9$
$x^2 + (y - 3)^2 = 25$
$x^2 + (y - 3)^2 = 5^2$
Это уравнение окружности с центром в точке $(0; 3)$ и радиусом $R=5$. Мы должны взять только ту часть этой окружности, которая удовлетворяет условию $y \ge 0$. Эта часть представляет собой дугу, расположенную над осью абсцисс и на самой оси.
Случай 2: $y < 0$
В этом случае $|y| = -y$, и уравнение принимает вид:
$x^2 + y^2 = 6(-y) + 16$
$x^2 + y^2 = -6y + 16$
Перенесем слагаемые с $y$ в левую часть и выделим полный квадрат:
$x^2 + y^2 + 6y = 16$
$x^2 + (y^2 + 6y + 9) = 16 + 9$
$x^2 + (y + 3)^2 = 25$
$x^2 + (y + 3)^2 = 5^2$
Это уравнение окружности с центром в точке $(0; -3)$ и радиусом $R=5$. Мы должны взять только ту часть этой окружности, которая удовлетворяет условию $y < 0$. Эта часть представляет собой дугу, расположенную под осью абсцисс.
Объединяя оба случая, получаем фигуру, состоящую из двух дуг окружностей. Обе окружности проходят через точки, где $y=0$:
$x^2 + (0 - 3)^2 = 25 \Rightarrow x^2 + 9 = 25 \Rightarrow x^2 = 16 \Rightarrow x = \pm 4$.
Таким образом, обе дуги соединяются в точках $(-4; 0)$ и $(4; 0)$.
Ответ: Множество точек является объединением двух дуг окружностей:
1. Дуги окружности $x^2 + (y - 3)^2 = 25$ при $y \ge 0$.
2. Дуги окружности $x^2 + (y + 3)^2 = 25$ при $y < 0$.
Эти дуги образуют замкнутую кривую, симметричную относительно обеих координатных осей.
Другие задания:
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