Question

Find the differential equation representing the family of circle touching the x-axis at origin

Answer 1

Answer:

Family of circles touching the axis at the origin. so, centre lies on y axis. let (0, a) be the centre of any member of member of the family. where we assume that a is arbitrary constant.

now, equation of circle is : x² + (y - a)² = a²

x² + y² + a² - 2ay = a²

x² + y² - 2ay = 0........(1)

hence, equation of family of circles is x² + y² - 2ay = 0

now, differentiate both sides with respect to x,

or, 2x + 2y.dy/dx - 2a.dy/dx = 0

or, x + y. dy/dx - a. dy/dx = 0

or, a = {x + y. dy/dx}/{dy/dx } ..........(2)

putting equation (2) in equation (1),

or, x² + y² - 2{x + y.dy/dx}/{dy/dx} × y = 0

or, (x² + y²) dy/dx - (2xy + 2y²dy/dx} = 0

or, (x² - y²)dy/dx - 2xy = 0

or, dy/dx = 2xy/(x² - y²) , which is required differential equation.