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Консультация № 145823
02.10.2008, 20:14
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Здравствуйте уважаемые эксперты, помогите пожалуйста с решением следующей задачи:
Составить уравнение линии, для каждой точки которой расстояние от точки A(0;1) вдвое меньше расстояния от прямой y=4
Составить уравнение линии, для каждой точки которой расстояние от точки A(0;1) вдвое меньше расстояния от прямой y=4
Обсуждение
03.10.2008, 07:39
общий
это ответ
Здравствуйте, Колос Алексей Юрьевич!
Решение.
Пусть точка M(x; y) принадлежит искомой линии. Тогда расстояние от нее до точки A(0; 1) равно ((x - 0)^2 + (y - 1)^2)^(1/2) (по теореме Пифагора), а до прямой y = 4 равно y - 4. Согласно условию задачи,
2((x - 0)^2 + (y - 1)^2)^(1/2) = y - 4. (*)
Уравнение (*) является уравнением искомой линии. Можно выполнить еще слеующие преобразования:
4((x - 0)^2 + (y - 1)^2) = (y - 4)^2,
4(x^2 + y^2 - 2y + 1) = y^2 - 8y + 16,
4x^2 + 4y^2 - 8y + 4 = y^2 - 8y + 16,
4x^2 + 3y^2 = 12,
(x^2)/3 + (y^2)/4 = 1. (**)
Уравнение (**) задает эллипс (замкнутую овальную кривую), все точки которого лежат внутри прямоугольника, образованного прямыми x = +-sqrt 3, y = +-2, а точка пересечения полуосей находится в центре координат.
Ответ: (x^2)/3 + (y^2)/4 = 1.
Решение.
Пусть точка M(x; y) принадлежит искомой линии. Тогда расстояние от нее до точки A(0; 1) равно ((x - 0)^2 + (y - 1)^2)^(1/2) (по теореме Пифагора), а до прямой y = 4 равно y - 4. Согласно условию задачи,
2((x - 0)^2 + (y - 1)^2)^(1/2) = y - 4. (*)
Уравнение (*) является уравнением искомой линии. Можно выполнить еще слеующие преобразования:
4((x - 0)^2 + (y - 1)^2) = (y - 4)^2,
4(x^2 + y^2 - 2y + 1) = y^2 - 8y + 16,
4x^2 + 4y^2 - 8y + 4 = y^2 - 8y + 16,
4x^2 + 3y^2 = 12,
(x^2)/3 + (y^2)/4 = 1. (**)
Уравнение (**) задает эллипс (замкнутую овальную кривую), все точки которого лежат внутри прямоугольника, образованного прямыми x = +-sqrt 3, y = +-2, а точка пересечения полуосей находится в центре координат.
Ответ: (x^2)/3 + (y^2)/4 = 1.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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