Question

If \( S \) and \( S' \) are the foci of the ellipse \( \frac{x^2}{18} + \frac{y^2}{9} = 1 \), and \( P \) is a point on the ellipse, then \( \min(\vec{SP} \cdot \vec{S'P}) + \max(\vec{SP} \cdot \vec{S'P}) \) is equal to:

• A. \( 3(1 + \sqrt{2}) \)
• B. \( 3(6 + \sqrt{2}) \)
• C. 9
• D. 27

Hint:

Use ellipse identities involving eccentricity and parametric coordinates for dot product evaluations.

Answer 1

The Correct Option is D

We use geometric properties of the ellipse and maximum/minimum dot product identities. 
Sum of maximum and minimum values of \( \vec{SP} \cdot \vec{S'P} \) gives: \[ \min + \max = 27 \]