Let be a parabola with vertex and directrix . Let an ellipse , of eccentricity pass through the focu

Q: JEE Main 2024 Conic Sections: Let be a parabola with vertex and directrix . Let an ellipse , of eccentricity pass through the focu...

The correct answer is A. This problem requires a strong understanding of the geometric properties of both parabolas and ellipses. We'll first determine the focus of the parabola using its vertex and directrix. This focus will then serve as a point on the given ellipse, which, combined with its eccentricity, will allow us to find its parameters and ultimately the square of the length of its latus rectum. --- **1. Key Concepts and Formulas** * **Parabola:** * The **vertex** of a parabola ( ) is the midpoint of the segment

Question

Let be a parabola with vertex and directrix . Let an ellipse , of eccentricity pass through the focus of the parabola . Then, the square of the length of the latus rectum of , is

Accepted Answer

This problem requires a strong understanding of the geometric properties of both parabolas and ellipses. We'll first determine the focus of the parabola using its vertex and directrix. This focus will then serve as a point on the given ellipse, which, combined with its eccentricity, will allow us to find its parameters and ultimately the square of the length of its latus rectum.

1. Key Concepts and Formulas

Parabola:
- The vertex of a parabola ( $V$ ) is the midpoint of the segment connecting its focus ( $F$ ) and the point where the axis of the parabola intersects the directrix ( $M_D$ ).
- The axis of the parabola is a line passing through the vertex and focus,

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