# $AB$ is a chord of a circle $C$. Let there be another point $P$ on the circumference of the circle, optimize $PA.PB$ and $PA+PB$

$AB$ is a chord of a circle $C$.
(a) Find a point $P$ on the circumference of $C$ such that $PA.PB$ is the maximum.
(b) Find a point $P$ on the circumference of $C$ which maximizes $PA+PB$.

My work:
(a)I draw a chord $AB$ on the cirlce $C$, and choose any random point $P$ to construct the triangle. Now, let the area of the triangle inscribed in the circle be $\Delta$ and the radius of the circle be $R$.
We have, $\Delta=PA.PB.\sin P$. So, $PA.PB=\dfrac{\Delta}{\sin P}$. So, $PA.PB$ attains its maximum value when $\sin P$ is lowest or $\angle P$ is lowest. But, is it all that can be derived? Please help me to proceed.

(b)$PA=2R\sin B,PB=2R\sin A$. So, $PA+PB=2R(\sin A+\sin B)$. Now, I have to maximize $\sin A+\sin B$ under restrictions. My intuition suggests, it is the maximum value when $\angle A=\dfrac{\pi}{3},\angle B=\dfrac{\pi}{3}$. Please help!

#### Solutions Collecting From Web of "$AB$ is a chord of a circle $C$. Let there be another point $P$ on the circumference of the circle, optimize $PA.PB$ and $PA+PB$"

While you are (almost) right that $\Delta=\frac12 = PA\cdot PB\cdot \sin P$, note that $\angle P$ is constant on each of the two arcs given by $A,B$! Thus maximizing $PA\cdot PB$ is equivalent to maximizing $\Delta$, equivalently to maximizing the height orthogonal to $AB$.