Question

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

• A. $y^2=x^2+2xy\frac{dy}{dx}$
• B. $y^2=x^2-2xy\frac{dy}{dx}$
• C. $x^2=y^2+xy\frac{dy}{dx}$
• D. $x^2=y^2+3xy\frac{dy}{dx}$

Answer 1

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

General equation of all such circles is ${(x-h)}^2+{(y-0)}^2=h^2$ (i)
where h is parameter
$\Rightarrow$ ${(x-h)}^2+y^2=h^2$ (ii)
Differentiating, we get $2(x-h)+2y\frac{dy}{dx}=0$
$h=x+y\frac{dy}{dx}$ to eliminate h, putting value of h in equation (i),
$\therefore$ we get $y^2=x^2+2xy\frac{dy}{dx}$