Question

If $3x+y=0$ is a tangent to the circle which has its centre at the point $(2,-1)$, then find the equation of the other tangent to the circle from the origin.

Answer 1

Using $p=r$, we get $r=\sqrt{\left(5/2\right)}$.
Any line through origin is $y=mx$.
If it is a tangent to $(2,-1),\sqrt{\left(5/2\right)}$, then using $p=r$,
we get
$\frac{2m+1}{\sqrt{(m^2+1)}}=\sqrt{\frac{5}{2}}$. Square
$2(4m^2+4m+1)=5(m^2+1)$
or $3m^2+8m-3=0$
or $(m+3)(3m-1)=0$
$\therefore m=-3,1/3$ where $-3$ corresponds to given tangent.
Hence the other tangent is $y=\frac{1}{3}x$ or $x-3y=0$.