The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and , respecti

Q: JEE Main 2024 Conic Sections: The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and , respecti...

The correct answer is 2. This solution aims to provide a detailed, step-by-step approach to solving the given hyperbola problem. We will systematically use the properties of hyperbolas, tangent conditions, and focal distances to arrive at the final answer. --- **1. Understanding the Hyperbola and its Standard Equations** A hyperbola centered at the origin with its transverse axis along the x-axis has the standard equation: where is the semi-transverse axis length and is the semi-conjugate axis length. Key formulas for s

Question

The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and , respectively. Let the line touch this hyperbola at . If is the product of the focal distances of the point , then is equal to .

Accepted Answer

This solution aims to provide a detailed, step-by-step approach to solving the given hyperbola problem. We will systematically use the properties of hyperbolas, tangent conditions, and focal distances to arrive at the final answer.

1. Understanding the Hyperbola and its Standard Equations

A hyperbola centered at the origin with its transverse axis along the x-axis has the standard equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ where $a$ is the semi-transverse axis length and $b$ is the semi-conjugate axis length. Key formulas for such a hyperbola are:

Eccentricity $e$ : $b^2 = a

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