Question

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

• A. $2xy\frac{dy}{dx}=x^2-y^2$
• B. $2xy\frac{dy}{dx}=y^2-x^2$
• C. $2xy\frac{dy}{dx}=x^2+y^2$
• D. $2xy\frac{dy}{dx}+x^2+y^2=0$

Answer 1

The correct option is B $2xy\frac{dy}{dx}=y^2-x^2$

Let the equation of family of circle passing through origin and having centre $(a,0)$ be

$(x-a{)}^2+(y-0{)}^2=a^2$

or, $x^2+a^2-2ax+y^2=a^2$

or, $x^2-2ax+y^2=\mathrm{0........}\left(i\right)$

On integrating equation (i), we get

or, $2x+2y\frac{dy}{dx}-2a=0$

or, $x+y\frac{dy}{dx}-a=0$

or, $x+y\frac{dy}{dx}-\left[\frac{x^2+y^2}{2x}\right]=0$

or, $2x^2+2xy\frac{dy}{dx}-x^2-y^2=0$

or, $2xy\frac{dy}{dx}+x^2-y^2=0$

or, $2xy\frac{dy}{dx}=y^2-x^2$

Hence, B is the correct option.