Question

The differential equation of the family of circles touching $y$-axis at the origin is

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

Answer 1

Answer: (B)

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
$(x-h{)}^2+(y-0{)}^2=h^2$
$\Rightarrow x^2+h^2-2hx+y^2=h^2$
$\Rightarrow x^2-2hx+y^2=0\dots (i)$
On differentiating both sides w.r.t x, we get
$2x-2h+2y\frac{dy}{dx}=0$
$\Rightarrow h=x+y\frac{dy}{dx}$
On putting $h=x+y\frac{dy}{dx}$ in Eq. (i), we get
$x^2-2\left(x+y\frac{dy}{dx}\right)x+y^2=0$
$\Rightarrow -x^2+y^2-2xy\frac{dy}{dx}=0$
$\Rightarrow \left(x^2-y^2\right)+2xy\frac{dy}{dx}=0$