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Консультация № 185782
08.04.2012, 12:34
67.46 руб.
0
3
1
Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Найти кривую, у которой нормаль в любой её точке равна расстоянию этой точки от начала координат.
Найти кривую, у которой нормаль в любой её точке равна расстоянию этой точки от начала координат.
Обсуждение
08.04.2012, 17:02
общий
Адресаты:
Мне эта задача сразу показалась какой-то странной. Взял я её не с учебника, переписал с листочка, который мне принесли с задачами, чтобы отправить сюда. Скажите, пожалуйста, эту задачу можно как-то переформулировать, чтобы она стала решаемой?
08.04.2012, 18:21
общий
это ответ
Здравствуйте, Aleksandrkib!
В задаче требуется найти такую кривую, для всякой точки которой отрезок нормали, заключённый между этой точкой и центром кривизны, равен расстоянию от этой точки до начала координат. Очевидно, что окружность с центром в начале координат удовлетворяет этому условию. Следовательно, искомое уравнение суть
Интуиция подсказывает, однако, что "подозрительной" является и равносторонняя гипербола.
Можно составить дифференциальное уравнение и решить его. Получим
Значит, помимо окружности, условию задачи соответствуют и равносторонние гиперболы, уравнения которых
С уважением.
В задаче требуется найти такую кривую, для всякой точки которой отрезок нормали, заключённый между этой точкой и центром кривизны, равен расстоянию от этой точки до начала координат. Очевидно, что окружность с центром в начале координат удовлетворяет этому условию. Следовательно, искомое уравнение суть
Интуиция подсказывает, однако, что "подозрительной" является и равносторонняя гипербола.
Можно составить дифференциальное уравнение и решить его. Получим
Значит, помимо окружности, условию задачи соответствуют и равносторонние гиперболы, уравнения которых
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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