Question

From a point $P\left(3,3\right)$ on the circle $x^{2}+y^{2}=18,$ two chords $PQ$ and $PR$ each of $2$ units length are drawn on this circle. The value of cos $\left(\angle Q P R\right)$ is equal to

• A. $\frac{1}{3 \sqrt{2}}$
• B. $-\frac{8}{9}$
• C. $\frac{\sqrt{2}}{3}$
• D. $\frac{- 4}{9}$

Answer 1

Answer: (B)

Let angle $OPR=\theta $
then in $\Delta OPR,$
$\cos\theta =\frac{2^{2} + \left(3 \sqrt{2}\right)^{2} - \left(3 \sqrt{2}\right)^{2}}{2 \times 2 \times 3 \sqrt{2}}$
$\Rightarrow \cos\theta =\frac{1}{3 \sqrt{2}}$
Now, $\cos\left(\angle Q P R\right)=\cos2\theta =2\cos^{2}\theta -1=2\left(\frac{1}{3 \sqrt{2}}\right)^{2}-1=\frac{1}{9}-1=-\frac{8}{9}$