Question

Obtain the differential equation of the family of circles having centre $(0,6)$ or centre lying on y-axis passing through the points $(a,0)$ and $(-a,0)$.

Answer 1

Let family of circle be

$(x{)}^2+(y-k{)}^2=r^2$ (enter $(o,k)$)

Passes through $(a,o)$

$a^2+k^2=r^2$

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

Differentiate implicitly

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

$\frac{dy}{dx}=\frac{-x}{y-k}$

$\frac{dy}{dx}+\frac{x}{y-k}=0$