Question

The differential equation of the family of circles whose center lies on $x-$axis and passing through origin is

• A. $x^2+y^2+\frac{dy}{dx}=0$
• B. $(y^2-x^2)dx-2xydy=0$
• C. $y^{2}dx+(x^2+2xy)dy=0$
• D. $xdy+ydx+x^{2}dx+y^{2}dy=0$

Answer 1

The correct option is B $(y^2-x^2)dx-2xydy=0$

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

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

$\Rightarrow 2x^2+2xy\frac{dy}{dx}=x^2+y^2$
Re-arranging this, we get

$\Rightarrow (y^2-x^2)dx-2xydy=0$
which is the required differential equation.

Hence, option B.