Question

Find the differential equation of the family of circles passing through the origin and having their centres on the x-axis

Answer 1

Given the circles centre on x-axis and the circle is passing through the origin.

Let it be (r, 0) and its radius r

Equation of the circle is

(x – a)² + (y – b)² = r²

(x – r)² + (y – 0)² = r²

x² – 2xr + r² + y² = r²

x² – 2xr + y² = r² – r²

x² – 2xr + y² = 0  ........(1)

Differentiating equation (1) with respect to ‘x’, we get

2x – 2r + 2y `("d"y)/("d"x)` = 0 dx

2x + 2y `("d"y)/("d"x)` = 2r

`x + y ("d"y)/("d"x)` = r

Substituting r value in equation (1), we get

`x^2 - 2x(x + y ("d"y)/("d"x)) + y^2` = 0

`x^2 - 2x^2 - 2xy  ("d"y)/("d"x) + y^2` = 0

`- x^2 - 2xy ("d"y)/("d"x) + y^2` = 0

Multiply by '_', we et

`x^2+ 2xy ("d"y)/("d"x) - y^2` = 0

Which is a required differential equation.