Question

Let $P$ and $Q$ be two distinct points on a circle which has centre at $C(2,3)$ and which passes through origin $O.$ If $OC$ is perpendicular to both the line segments $CP$ and $CQ$, then the set $\{P,Q\}$ is equal to

• A. $\left\{\right(2+2\sqrt{2},3+\sqrt{5}),(2-2\sqrt{2},3-\sqrt{5}\left)\right\}$
• B. $\left\{\right(4,0),(0,6\left)\right\}$
• C. $\left\{\right(-1,5),(5,1\left)\right\}$
• D. $\left\{\right(2+2\sqrt{2},3-\sqrt{5}),(2-2\sqrt{2},3+\sqrt{5}\left)\right\}$

Answer 1

The correct option is C $\left\{\right(-1,5),(5,1\left)\right\}$


Equation of circle : $(x-2{)}^2+(y-3{)}^2=13$
$m_1=\frac{3}{2}$
$\Rightarrow m_2=\frac{-2}{3}=\tan \theta$
$\therefore \sin \theta =\frac{2}{\sqrt{13}},\cos \theta =\frac{-3}{\sqrt{13}}$

Coordinates of $P,Q$
$(2\pm \sqrt{13}\cos \theta ,3\pm \sqrt{13}\sin \theta )$
$=(2\pm \sqrt{13}(-\frac{3}{\sqrt{1}3}),3\pm \sqrt{13}\left(\frac{2}{\sqrt{1}3}\right))$
$=(-1,5),(5,1)$