Question

Find the differential equation of family of circles of all of radius 5, with their centres on the y-axis.

Answer 1

General equation of circle is

${(x-a)}^2+{(y-b)}^2=r^2$

Given centre is on y-axis and radius $=5$units

$\therefore$ centre $=(0,b)$

$\therefore$ our equation becomes

${(x-0)}^2+{(y-b)}^2={\left(s\right)}^2$

$x^2+{(y-b)}^2=2s......\left(i\right)$

Differentiating w.r.t $x$

$2x+2(y-b)[\frac{dy}{dx}-0]=0$

$\Rightarrow$ $(y-b)=\frac{-x}{\left(\frac{dy}{dx}\right)}$

Substituting $(y-b)$ in (i)

$x^2+\frac{x^2}{{\left(\frac{dy}{dx}\right)}^2}=25$

$\Rightarrow$ $x^2{\left(\frac{dy}{dx}\right)}^2+x^2=25{\left(\frac{dy}{dx}\right)}^2$

$\Rightarrow$ $x^2{\left(\frac{dy}{dx}\right)}^2-25{\left(\frac{dy}{dx}\right)}^2+x^2=0$

$\Rightarrow$ ${\left(\frac{dy}{dx}\right)}^2[x^2-25]+x^2=0$