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Консультация № 148594
27.10.2008, 09:38
0.00 руб.
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1
1
Уважаемые эксперты, расщитываю только на вашу помощь, помогите справиться с заданием:
Найти длину наибольшего отрезка, который принадлежит множеству D:
D = { (x;y) : (|x-1| - 2)^2 + y^2 <=9}.
Заранее огромное спасибо.
Найти длину наибольшего отрезка, который принадлежит множеству D:
D = { (x;y) : (|x-1| - 2)^2 + y^2 <=9}.
Заранее огромное спасибо.
Обсуждение
28.10.2008, 21:10
общий
это ответ
Здравствуйте, Kafka!
Решение.
При x > 1 заданное неравенство тождественно неравенству
(x – 3)^2 + y^2 ≤ 9,
задающему круг, центр которого находится в точке (3; 0), а радиус равен 3.
При x < 1 заданное неравенство тождественно неравенству
(-x – 1)^2 + y^2 ≤ 9
или уравнению
(x + 1)^2 + y^2 ≤ 9,
задающему круг, центр которого находится в точке (-1; 0), а радиус равен 3.
При x = 1 заданное неравенство принимает вид
4 + y^2 ≤ 9, или
y^2 ≤ 5,
что равносильно двойному неравенству: -√5 ≤ y ≤ √5.
Таким образом, множество D представляет собой фигуру, ограниченную дугой окружности
(x – 3)^2 + y^2 = 9,
расположенной правее прямой x = 1, и дугой окружности
(x + 1)^2 + y^2 = 9,
расположенной левее прямой x = 1. Указанные дуги пересекаются в точках (1; -√5) и (1; √5).
Если выполнить соответствующий рисунок, то можно наглядно убедиться в том, что наибольшим будет отрезок AB, где A(-4; 0), B(6; 0). Следовательно, длина наибольшего отрезка равна
|AB| = 6 – (-4) = 10.
Ответ: 10.
С уважением.
Решение.
При x > 1 заданное неравенство тождественно неравенству
(x – 3)^2 + y^2 ≤ 9,
задающему круг, центр которого находится в точке (3; 0), а радиус равен 3.
При x < 1 заданное неравенство тождественно неравенству
(-x – 1)^2 + y^2 ≤ 9
или уравнению
(x + 1)^2 + y^2 ≤ 9,
задающему круг, центр которого находится в точке (-1; 0), а радиус равен 3.
При x = 1 заданное неравенство принимает вид
4 + y^2 ≤ 9, или
y^2 ≤ 5,
что равносильно двойному неравенству: -√5 ≤ y ≤ √5.
Таким образом, множество D представляет собой фигуру, ограниченную дугой окружности
(x – 3)^2 + y^2 = 9,
расположенной правее прямой x = 1, и дугой окружности
(x + 1)^2 + y^2 = 9,
расположенной левее прямой x = 1. Указанные дуги пересекаются в точках (1; -√5) и (1; √5).
Если выполнить соответствующий рисунок, то можно наглядно убедиться в том, что наибольшим будет отрезок AB, где A(-4; 0), B(6; 0). Следовательно, длина наибольшего отрезка равна
|AB| = 6 – (-4) = 10.
Ответ: 10.
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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