Discussion Forum

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

Answer:

Explanation:

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  ............(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

 $x^{2}-2\left( x+y \frac{dy}{dx}\right)x+y^{2}=0$

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

 $\Rightarrow$     $(x^{2}-y^{2})+2xy\frac{dy}{dx}=0$