Question

From the differential equation of the family of circles touching the y-axis at origin

Options

• `2xy  (dy)/(dx)`

• `2xy  (dy)/(dx) + x^2 - y^2` = 0

• `(dy)/(dx) - y^2` = 0

• None of these

Answer 1

`2xy  (dy)/(dx) + x^2 - y^2` = 0

Explanation:

The equation of the circle with centre (a, 0) and radius a, which touches y-axis at origin

`(x - a)^2 + y^2 = a^2` 

or `x^2 + y^2 = 2ax`  ......(1)

Differentiating w.r.t. x

`2x + 2yy^1 = 2a`

or `x + yy^1 = a`

Putting value of a in (1)

`x^2 + y^2 = 2x(x + yy^1) = 2x^2 + 2xyy^1`

∴ Required differential equation is `2xy  (dy)/(dx) + x^2 - y^2` = 0.