Question

The differential equation of all circles passing through the origin and having their centres on the X-axis, is

• A. $y^2=x^2+2xy\frac{dy}{dx}$
• B. $y^2=x^2-2xy\frac{dy}{dx}$
• C. $y^2=x^2+xy\frac{dy}{dx}$
• D. $y^2=x^2+3xy\frac{dy}{dx}$

Answer 1

Answer: (A)

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

The equation of the family of circles passing through the origin and having their centres on X-axis
${(x-a)}^2+{(y-0)}^2=a^2$
$x^2+y^2-2ax=\mathrm{0...}\left(i\right)$
On differentiating w.r.t $x$ we get
$2x+2y\frac{dy}{dx}-2a=0$
$\Rightarrow a=x+y\frac{dy}{dx}$
On substituting the value of $a$ in Eq.(i) we get
$x^2+y^2-2x^2-2xy\frac{dy}{dx}=0$
$\Rightarrow y^2=x^2+2xy\frac{dy}{dx}$
which is the required differential equation