Let the hyperbola pass through the point . A parabola is drawn whose focus is same as the focus of w

Q: JEE Main 2024 Conic Sections: Let the hyperbola pass through the point . A parabola is drawn whose focus is same as the focus of w...

The correct answer is A. This problem is a comprehensive test of your understanding of the fundamental properties of hyperbolas and parabolas, and your ability to synthesize information from different conic sections. We will systematically use the definitions of foci, directrix, eccentricity, and latus rectum for both curves to determine the equation of the parabola and then check which given point lies on it. --- **1. Prerequisites: Key Concepts and Formulas** Before we begin, let's review the essential properties of h

Question

Let the hyperbola pass through the point . A parabola is drawn whose focus is same as the focus of with positive abscissa and the directrix of the parabola passes through the other focus of . If the length of the latus rectum of the parabola is e times the length of the latus rectum of , where e is the eccentricity of H, then which of the following points lies on the parabola?

Accepted Answer

This problem is a comprehensive test of your understanding of the fundamental properties of hyperbolas and parabolas, and your ability to synthesize information from different conic sections. We will systematically use the definitions of foci, directrix, eccentricity, and latus rectum for both curves to determine the equation of the parabola and then check which given point lies on it.

1. Prerequisites: Key Concepts and Formulas

Before we begin, let's review the essential properties of hyperbolas and parabolas that we will apply:

Standard Hyperbola (Transverse axis along x-axis):
- Equation: $H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
- Foci: $(\pm ae, 0)$ , where $a$ is the semi-transverse axis and $e$ is the eccentricity.
- Eccentricity ( $e$ ): $e^2 = 1 + \frac{b^2}{a^2}$ (for hyperbola, $e > 1$ ).
- Length of Latus Rectum ( $\text{LLR}_H$ ): $\frac{2b^2}{a}$ .

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