Let P and Q be two distinct points on a circle which has centre C(2,3) and which passes th…

Question

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

Accepted Answer

$r=OC=\sqrt{13}$. $P,Q$ lie on the circle along the line through $C$ perpendicular to $OC=(2,3)$, at distance $r$ from $C$: unit perpendicular $\left(\mp\frac3{\sqrt{13}},\pm\frac2{\sqrt{13}}\right)$.

$$P=(2,3)+\sqrt{13}\left(-\frac3{\sqrt{13}},\frac2{\sqrt{13}}\right)=(-1,5),\qquad Q=(2,3)+\sqrt{13}\left(\frac3{\sqrt{13}},-\frac2{\sqrt{13}}\right)=(5,1).$$

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