Q. If $\left( a\cos\theta_{i},\ a\sin\theta_{i} \right),\ i = 1,\ 2,\ 3$ represents the vertices of an equilibrium triangle inscribed in a circle, then
A) $\cos\theta_{1} + \cos\theta_{2} + \cos\theta_{3} = 0$
B) $\sec\theta_{1} + \sec\theta_{2} + \sec\theta_{3} = 0$
C) $\tan\theta_{1} + \tan\theta_{2} + \tan\theta_{3} = 0$
D) $\cot\theta_{1} + \cot\theta_{2} + \cot\theta_{3} = 0$
Explanation
(a)
Given points lie on a circle $x^{2} + y^{2} = a^{2}$ and in case of an equilateral triangle centroid is same as circumcentre. Circumcentre of given triangle is at origin or centroid is at origin
$\frac{a\cos\theta_{1} + a\cos\theta_{2} + a\cos\theta_{3}}{3} = 0$,
and $\frac{a\sin\theta_{1} + a\sin\theta_{2} + a\sin\theta_{3}}{3} = 0$
$\sum_{}^{}{\cos\theta_{1}} = 0,\ \sum_{}^{}{\sin\theta_{1}} = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.