Question

The area of cyclic quadrilateral $PQRS$ is $4\sqrt{3}$ units. The radius of the circumcircle of $△PQR$ is $2$. If $PQ=2,QS=2\sqrt{3}$, then the value of product $QR$ and $RS$ is

• A. $12$
• B. $8$
• C. $8\sqrt{3}$
• D. None of these

Answer 1

Answer: (A), (B)

The correct options are
A $8$
B $12$

Area of $PQRS=4\sqrt{3}$ ...(1)

$\Rightarrow \frac{1}{2}(ad+bc)\sin P=4\sqrt{3}\Rightarrow (2d+bc)\sin P=8\sqrt{3}$ ....(2)

In $△PQR$, by sine formula, $\frac{2\sqrt{3}}{\sin P}=2R$;

$R=$ radius of circumcircle $△PQR$ which is same as that for $△PQR=2$ (given)

$\Rightarrow \sin P=\frac{\sqrt{3}}{2}\Rightarrow \cos P=\pm \frac{1}{2}$ ...(3)

$\Rightarrow \pm \frac{1}{2}=\frac{4+d^2-12}{2\left(2\right)\left(d\right)}\Rightarrow \pm 2d=d^2-8$

$\Rightarrow d^2\mp 2d-8=0\Rightarrow d=2,4$

$\therefore$ From (2) and (3) and (4);

$bc=16-2d=12,8$

$\Rightarrow \left(Qr\right).\left(SR\right)=12,8$