Question

Two points $O(0,0)$ and $A(3,\sqrt{3})$ along with another point $P$ form an equilateral triangle. Find the coordinate of $P$.

Answer 1

$OA=\sqrt{9+3}=\sqrt{12}$

and slope of $OA=\frac{\sqrt{3}}{3}=\frac{1}{\sqrt{3}}$

So, for $P$ slope$=\frac{-1}{\sqrt{3}}$

and distance$=\sqrt{12}$

Thus, point will be $P(3,-\sqrt{3})$

and in another case, slope of $PO=\tan (\frac{\pi }{6}+\frac{\pi }{3})$

slope$=\tan \left(\frac{\pi }{2}\right)=\infty$

So, $P$ lies on y-axis.

with distance$=\sqrt{12}$

Let $P\equiv (a,0)$

So, $\sqrt{a^2}=\sqrt{12}$

So, $a=\sqrt{12}$

So, $P\equiv (\sqrt{12},0)$