ISI2015-MMA-90

Question

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

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

Answer 1

The general equation of a circle having centre at $\left ( h,k \right )$ and radius $r$ is  :   $\left ( x-h \right )^{2}+\left ( y-k \right )^{2}=r^{2}$

Now, if the circle touches  $y-axis$  at the origin,  it's Centre would be lying at point   $\left ( a,0 \right )$  ($"a"$ is assumed)  and radius $a$ units.

So, the equation of circle would be:    $\left ( x-a \right )^{2}+\left ( y-0 \right )^{2}=a^{2}$

$\Rightarrow$   $x^{2}+y^{2}=2ax$  -----  $\left ( 1 \right )$

Differentiating on  both sides w.r.t   $x$, we get   

$2x+2y\frac{dy}{dx}=2k$    $\Rightarrow$  $a=x+y\frac{dy}{dx}$

Now, substituting the value of $a$ in  $\left ( 1 \right )$, we get

$x^{2}+y^{2}=2x \left ( x+y\frac{dy}{dx} \right )$

Rearranging the terms and simplifying, we get the differential equation as:   $x^{2}-y^{2}+2xy\frac{dy}{dx}=0$

Option D is the correct answer.