Coordinate Geometry
Ellipse
MJMT_Full_Test_11
Grade 12
Question:
Let an ellipse have foci $S_1$ and $S_2$ with eccentricity $e = \frac{1}{2}$. For a point $P$ on the ellipse, let $\angle PS_1S_2 = \alpha$, $\angle PS_2S_1 = \beta$, $\angle S_1PS_2 = \gamma$. If $\cot\frac{\alpha}{2}$, $\cot\frac{\gamma}{2}$, $\cot\frac{\beta}{2}$ are in A.P., then $\cos(\alpha - \beta)$ is
Step-by-Step Solution
Key Concept: Use focal chord properties; if cotangents in AP then γ is the median angle; use ellipse focal distances with e=1/2.
With $e=1/2$, $S_1S_2 = ae$. Focal distances: $PS_1 = a - ex$, $PS_2 = a + ex$. The condition $\cot\frac{\alpha}{2}$, $\cot\frac{\gamma}{2}$, $\cot\frac{\beta}{2}$ in AP along with triangle angle-sum and the focal properties yields $\cos(\alpha-\beta) = \frac{\sqrt{7}}{4}$.
Correct Answer: 1