Question

Let $P$ is any arbitrary point on the circumcircle of a given equilateral triangle $A B C$ of side length $\ell$ units, then $|\overrightarrow{P A}|^2+|\overrightarrow{P B}|^2+|\overrightarrow{P C}|^2$ is always equal to

• A. $2\ell ^{2}$
• B. $2\sqrt{3}\ell ^{2}$
• C. $\ell ^{2}$
• D. $3\ell ^{2}$

Answer 1

Answer: (A)

Let position vector of $P, A, B, C$ are $\vec{p}, \vec{a}, \vec{b}$ and $\vec{c} \& O(\vec{O})$ be the circumcentre of equilateral
triangle $A B C \Rightarrow|\vec{p}|=|\vec{a}|=|\vec{b}|=|\vec{c}|=\frac{\ell}{\sqrt{3}}$
$|\overrightarrow{P A}|^2=|\vec{a}-\vec{p}|^2=|\vec{a}|^2+|\vec{p}|^2-2 \vec{a} \cdot \vec{p}$
Now, $\sum \mid \overrightarrow{P A}\mid^2=6 \cdot \frac{\ell^2}{3}-2 \vec{p}(\vec{a}+\vec{b}+\vec{c})$ as
$\frac{\vec{a}+\vec{b}+\vec{c}}{3}=0$ hence $\sum|\overrightarrow{P A}|^2=2 \ell^2$