Question

Form the differential equation of the family of circles touching the X-axis at the origin.

Answer 1

Let $(0,a)$ be the co-ordinates of the centre, 'a' is an arbitrary constant.

Then, the equation is $x^2+(y-a{)}^2=a^2$

$\therefore x^2+y^2+a^2-2ay=a^2$

$x^2+y^2-2ay=0$ …. $\left(1\right)$

Different w.r.t. 'x' we get

$2x+2y+\frac{dy}{dx}-2a\frac{dy}{dx}=0$ …..$\left(2\right)$

Multiplying $\left(1\right)$ by $dy/dx$

$(x^2+y^2)\frac{dy}{dx}-2ay\frac{dy}{dx}=0$ …..$\left(3\right)$

Subtracting $\left(3\right)$ from $\left(2\right)$ we get

$2xy+(y^2-x^2)\frac{dy}{dx}=0$

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