Question:medium

The circumcenter of the equilateral triangle having the three points $\theta_1, \theta_2, \theta_3$ lying on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as its vertices is $(r,s)$. Then the average of $\cos(\theta_1-\theta_2), \cos(\theta_2-\theta_3)$ and $\cos(\theta_3-\theta_1)$ is

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For an equilateral triangle inscribed in an ellipse, the circumcenter and centroid coincide. This fact allows you to relate the coordinates of the center $(r,s)$ to the sum of the coordinates of the vertices, which in turn can be related to the eccentric angles.

Updated On: Mar 30, 2026

$\frac{1}{2}[\frac{3r^2}{a^2}+\frac{3s^2}{b^2}-1]$
$\frac{3}{2}[\frac{r^2}{a^2}+\frac{s^2}{b^2}]$
$\frac{1}{3}[\frac{r^2}{a^2}+\frac{s^2}{b^2}]$
$\frac{1}{3}[\frac{r^2}{a^2}+\frac{s^2}{b^2}+\frac{rs}{ab}]$

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The Correct Option is A

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