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.

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