Question

The equation of tangents drawn from the origin to the circle $x^2+y^2-2rx-2hy+h^2=0$ are $x=0$ and $(h^2-r^2)x-2rhy=0$.

Answer 1

The centre is $(r,h)$ and radius is $r$. Now any line through origin is $y-mx=0$. Apply the condition of tangency, i.e., $p=r$
$\frac{h-mr}{\sqrt{(1+m^2)}}=r$
or $h^2+m^2{r}^2-2mhr=r^2+m^2{r}^2$
or $0m^2+2mhr+(r^2-h^2)=0$
$\therefore m=\infty ,m=(h^2-r^2)/2hr$.
Putting the values of $m$ in $y/x=m$
Tangents are $x=0$ for $m=\infty$,
and $(h^2-r^2)x-2rhy=0$.