Question

The differential equation of all circles touching the axis of $ y $ at origin and centre on the $ x $ -axis is given by

• A. $ xy \, \frac{dy}{dx}-x^{2}+y^{2} = 0 $
• B. $ 2xy \, \frac{dy}{dx}-x^{2}-y^{2} = 0 $
• C. $ \left(x^{2}+y^{2}\right) \, \frac{dy}{dx}-2xy = 0 $
• D. None of the above

Answer 1

Answer: (D)

The equation of circle touching the $y$ -axis at $(0,0)$ and centre lies on $x$ -axis is $x^{2}+y^{2}-2 g x=0$
Let the required equation of circle be
$x^{2}+y^{2}-2 g x=0\,\,\,\,...(i)$
On differentiating w.r.t. $x$, we get
$2 x+2 y \frac{d y}{d x}-2 g=0$
$\Rightarrow y \frac{d y}{d x}+x-\frac{x^{2}+y^{2}}{2 x}=0 \,\,\,[$ from Eq. (i) $]$
$\Rightarrow 2 x y \frac{d y}{d x}+x^{2}-y^{2}=0$