Question

Equation of a tangent to the circle with centre $(2,-1)$ is $3x+y=0$, then the twice of square of the length of the tangent to the circle from the point $(23,17)$ is

Answer 1

Given center of circle as $(2,-1)$.

Equation of tangent to the circle is $3x+y=0$

Radius =Length of the perpendicular from the centre $(2,-1)$ on the tangent $3x+y=0$ is

$\left|\frac{6-1}{\sqrt{9+1}}\right|=\frac{5}{\sqrt{10}}$

So, $r=\frac{5}{\sqrt{10}}$

.Equation of the circle is

$(x-2{)}^2+(y+1{)}^2=\frac{5}{2}$

$\Rightarrow \left[\right(x-2{)}^2+(y+1{)}^2]-\frac{5}{2}=0$

The required length $=\left[\right(23-2{)}^2+(17+1{)}^2]-\frac{5}{2}$

$=[441+324]-\frac{5}{2}=\frac{1525}{2}$

Twice of length is $1525$