Let the foci and length of the latus rectum of an ellipse and , respectively. Then, the square of th

Q: JEE Main 2024 Conic Sections: Let the foci and length of the latus rectum of an ellipse and , respectively. Then, the square of th...

The correct answer is 5. This problem requires us to first extract parameters from a given ellipse and then use these parameters to determine the eccentricity of a related hyperbola. We will use the standard definitions and formulas for ellipses and hyperbolas. ### **1. Key Concepts and Formulas** Before diving into the solution, let's recall the essential properties of ellipses and hyperbolas that will be used: * **Ellipse (Major axis along x-axis):** The standard equation is , where . * Foci: , where is the eccentrici

Question

Let the foci and length of the latus rectum of an ellipse and , respectively. Then, the square of the eccentricity of the hyperbola equals

Accepted Answer

This problem requires us to first extract parameters from a given ellipse and then use these parameters to determine the eccentricity of a related hyperbola. We will use the standard definitions and formulas for ellipses and hyperbolas.

1. Key Concepts and Formulas

Before diving into the solution, let's recall the essential properties of ellipses and hyperbolas that will be used:

Ellipse (Major axis along x-axis): The standard equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , where $a>b$ $a > b$ .
- Foci: $(\pm ae, 0)$ , where $e$ is the eccentricity.
- Length of the latus rectum: $\frac{

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Solution

1. Key Concepts and Formulas

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