If $\mathrm{P}\left(\frac{\pi}{3}\right)$ and $\mathrm{Q}\left(\frac{2 \pi}{3}\right)$ represent two points…
If $\mathrm{P}\left(\frac{\pi}{3}\right)$ and $\mathrm{Q}\left(\frac{2 \pi}{3}\right)$ represent two points on the circle $x^2+y^2-4 x+6 y-12=0$ in parametric form, then the length of the chord PQ is
$4 \sqrt{3}$
$5$
$5 \sqrt{2}$
$13$
Solution
Given equation of circle
$\begin{aligned}
& x^2+y^2-4 x+6 y-12=0 \\
& \Rightarrow(x-2)^2+(y+3)^2=5^2
\end{aligned}$
so parametric coordinate $\equiv 2+5 \cos \theta,-3+5 \sin \theta$ )
Now, for point $P, P \equiv\left(2+5 \cos \left(\frac{\pi}{3}\right),-3+5 \sin \left(\frac{\pi}{3}\right)\right)$
$P \equiv\left(\frac{9}{2},-3+\frac{5 \sqrt{3}}{2}\right)$
and for point $Q, Q \equiv\left(2+5 \cos \left(\frac{2 \pi}{3}\right),-3+5 \sin \left(\frac{2 \pi}{3}\right)\right)$
$Q \equiv\left(\frac{-1}{2},-3+\frac{5 \sqrt{3}}{2}\right)$
Now, $P Q=\sqrt{\left(\frac{10}{2}\right)^2+0^2} \equiv 5$