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 $x^2+y^2-2gx=0$
Let the required equation of circle be
$x^2+y^2-2gx=0...(i)$
On differentiating w.r.t. $x$, we get
$2x+2y\frac{dy}{dx}-2g=0$
$\Rightarrow y\frac{dy}{dx}+x-\frac{x^2+y^2}{2x}=0[$ from Eq. (i) $]$
$\Rightarrow 2xy\frac{dy}{dx}+x^2-y^2=0$