Question

The differential equation of all circles passing through the origin and having their centers on the x-axis is:

• A. $y^2=x^2+2xy\frac{dy}{dx}$
• B. $y^2=x^2-2xy\frac{dy}{dx}$
• C. $x^2=y^2+2xy\frac{dy}{dx}$
• D. None of these

Answer 1

The correct option is A $y^2=x^2+2xy\frac{dy}{dx}$

General equation of a circle can be given as
$x^2+y^2+2gx+2fy+c=0$
Since, it is given that this circle passes through origin and centre lies on $x-$axis

$\therefore$ Center of circle is $(-g,0)$

So, the equation of a circle passing through origin and centre on $x-$axis is
$x^2+y^2+2gx=0$ ....(1)

$\Rightarrow g=-\frac{x^2+y^2}{2x}$

Differentiating (1) w.r.t $x$
$2x+2y\frac{dy}{dx}+2g=0$ ....(2)
Substituting $g$ in (2), we get
$\Rightarrow 2x+2y\frac{dy}{dx}-\frac{x^2+y^2}{x}=0$

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

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

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