Question

Let $A$ be the centre of the circle $x^2+y^2-2x-4y-20=0.$ Suppose that the tangents at the points $B(1,7)$ and $D(4,-2)$ on the circle meet at point $C$. Then the area of the quadrilateral $ABCD$ is

Answer 1

Centre of the circle is $A(1,2)$ and radius is $\sqrt{1^2+2^2+20}=5$
$AB$ is a line segment parallel to $y$-axis and therefore tangent at $B$ will be parallel to $x$-axis.
Let $C\equiv (a,7)$
Using $DC=BC$, we get $C\equiv (16,7)$
$\Rightarrow BC=15$

So, area $\left(△ABC\right)=\frac{1}{2}\times AB\times BC=\frac{75}{2}$
$\Rightarrow$ Area of quadrilateral $=75$