Mathematics | MHT CET 2016 | Discussion Page

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

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

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

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

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

Option 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 ............(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

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

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

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

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