Question

Find the area of the quadrilateral formed by a pair of tangents from the point $(4,5)$ to the circle $x^2+y^2-4x-2y-11=0$ and $a$ pair of its radii.

Answer 1

Now,

$x^2+y^2-4x-2y-11=0{\left(x-2\right)}^2+{\left(y-1\right)}^2=16={\left(4\right)}^2$

$Area\left(ABCD\right)=2\times \frac{1}{2}\times AB\times BC=AB\times BC=2\times 4=8c{m}^{2}BC=4AB=\sqrt{\left(16\right)+\left(25\right)-4\left(4\right)-2\left(5\right)-11}=\sqrt{30-10-16}=\sqrt{4}=2$

Hence, this is the answer.