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

Options

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

Answer

B

Explanation

Eq. of family of circles passing through the origin & having centres on the x-axis is :
` x^(2) +y^(2) +2gx =0`
` 2x +2y.(dy)/(dx) +2g =0`
` Rightarrrow g=- [x +y(dy)/(dx)]`
Putting the values of (g) in eq. (1) we get
` 2xy (dy)/(dx) = y^(2) -x^(2)`