Question

The area of the quadrilaterial formed by the two tangents from $(4,5)$ to the circle $x^2+y^2-4x-2y-11=0$ and a pair of radii is

• A. $2$
• B. $4$
• C. $8$
• D. $16$

Answer 1

The correct option is C $8$

$x^2+y^2-4x-2y-11=0$
Radius $=\sqrt{4+1+11}=4$
$\therefore$ Length of tangent from $P(4,5)$ to $x^2+y^2-4x-2y-11=0$ is
$P{T}_2=P{T}_1=\sqrt{4^2+5^2-4\times 4-2\times 5-11}$
$=\sqrt{4}$
$=2$
$\therefore$ area of quadrilateral $P{T}_{2}C{T}_1$
$=$ area of $P{T}_{2}C+P{T}_{1}C.$

$=\frac{1}{2}\times P{T}_2\times r+\frac{1}{2}\times P{T}_1\times r$

$=\frac{1}{2}\times r\left[2P{T}_1\right]$

$=r\times P{T}_1$

$=4\times 2$

$=8$ sq. units.