Question

Let $x^2+y^2-4x-2y-11=0$ be a circle. A pair of tangents from the point (4, 5) with a pair of radii form a quadrilateral of area ___.

Answer 1

The equation of the circle is $x^2+y^2-4x-2y-11=0$.
Its center is (2, 1) and radius is $\sqrt{4+1+11}=4=BC$.

$AC=2\sqrt{5}...\left(FromDistanceformula\right)\therefore AB=2...\left(Fromright\Delta ABC\right)\therefore AreaofquadrilateralABCD=2\times \left(Areaof\Delta ABC\right)=2\times \frac{1}{2}\times AB\times BC=2\times \frac{1}{2}\times 2\times 4=8sq.units$