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 $\left(\angle QPR\right)$ 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, $\cos \left(\angle QPR\right)=\cos 2\theta =2{\cos }^2\theta -1=2{\left(\frac{1}{3\sqrt{2}}\right)}^2-1=\frac{1}{9}-1=-\frac{8}{9}$