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

Options

A. `(x^(2)+y^(2))(dy)/(dx)-2xy=0`
B. `(x^(2)-y^(2))+2xy(dy)/(dx)=0`
C. ` (x^(2)-y^(2)) (dy)/(dx) - 2xy = 0`
D. `(x^(2)+y^(2))(dy)/(dx)+2xy=0`

Answer

b

Explanation

Let centre of circle on X - axis be (h,0)
The radius of circle will be h .

` :.` The equation of circle having centre (h,0) and radius h is
` (x-h)^(2)+ (y-0)^(2) = h^(2)`
` rArr x^(2) +h^(2) - 2hx + y^(2) = h^(2)`
` rArr x^(2) - 2hx +y^(2) = 0`
On differentiating both sides w.r.t x, we get
` 2x - 2h +2y(dy)/(dx) = 0 rArr h = x+y (dy)/(dx)`
On putting `h = x+y (dy)/(dx)` in Eq. (i), we get
` x^(2) - 2 ( x + y (dy)/(dx)) x +y^(2) = 0`
` rArr -x^(2) +y^(2) - 2xy (dy)/(dx) =0`
` rArr (x^(2) -y^(2)) +2xy (dy)/(dx) =0`
` rArr (x^(2) -y^(2)) +2xy (dy)/(dx)=0 `