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Консультация № 198610
17.05.2020, 23:14
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Найти ось симметрии и вершину x^2 + 4xy + 4y^2 - 6x - 2y + 1 = 0
Заранее спасибо.
Найти ось симметрии и вершину x^2 + 4xy + 4y^2 - 6x - 2y + 1 = 0
Заранее спасибо.
Обсуждение
24.05.2020, 19:05
общий
это ответ
Здравствуйте, moonfox!
Из условия задачи следует, что приведённое уравнение является "неканоническим" уравнением параболы. Для приведения его к каноническому виду найдём угол поворота при переходе к новой системе координат, который для уравнения вида
определяется соотношениями
или
В данном случае для A = 1, B = 2 и C = 4 имеем
откуда
и
Тогда переход к новым координатам (x', y') осуществляется по формулам
Подставляя эти значения в исходное уравнение, получаем
Это уравнение параболы с вершиной в точке x' = 0, y' = 1/[$8730$]5 и осью симметрии, параллельной оси Ox' (в новой системе координат). В старой системе координаты вершины будут равны
а ось симметрии будет проходить через вершину (1/5, 2/5) под углом [$945$] к оси Ox, то есть её уравнение будет иметь вид
или
Из условия задачи следует, что приведённое уравнение является "неканоническим" уравнением параболы. Для приведения его к каноническому виду найдём угол поворота при переходе к новой системе координат, который для уравнения вида
определяется соотношениями
или
В данном случае для A = 1, B = 2 и C = 4 имеем
откуда
и
Тогда переход к новым координатам (x', y') осуществляется по формулам
Подставляя эти значения в исходное уравнение, получаем
Это уравнение параболы с вершиной в точке x' = 0, y' = 1/[$8730$]5 и осью симметрии, параллельной оси Ox' (в новой системе координат). В старой системе координаты вершины будут равны
а ось симметрии будет проходить через вершину (1/5, 2/5) под углом [$945$] к оси Ox, то есть её уравнение будет иметь вид
или
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