The differential equation for the family of curve `x^2+y^2-2a y=0,` where `a` is an arbitrary constant, is

Options

A. `2(x^(2)-y^(2))y^(')=xy`
B. `2(x^(2)+y^(2))y^(')=xy`
C. `(x^(2)-y^(2))y^(')=2xy`
D. `(x^(2)+y^(2))y^(')=2xy`

Answer

C

Explanation

The given family of curves is `x^(2)+y^(2)-2ay=0` ………….(1)
Differentiating w.r.t x, we get `2x+2y(dy)/(dx)-2a(dy)/(dx)=0`
or `2x+2y(dy)/(dx) -(x^(2)+y^(2))/(y)(dy)/(dx)=0` [Using equation (1)]
or `2xy+(2y^(2)-x^(2)-y^(2))y^(')=0`
or `(x^(2)-y^(2))y^(')=2xy`