A hyperbola whose transverse axis is along the major axis of the conic, and has vertices at the foci

Q: JEE Main 2023 Conic Sections: A hyperbola whose transverse axis is along the major axis of the conic, and has vertices at the foci...

The correct answer is A. This problem involves understanding the properties of both ellipses and hyperbolas, specifically their standard forms, how to determine their axes, foci, vertices, and eccentricity. The key is to carefully extract information from the given ellipse and use it to construct the equation of the hyperbola. **Key Concepts and Formulas:** 1. **Standard Equation of an Ellipse centered at the origin:** * If the major axis is along the x-axis ( ): * Vertices: * Foci: * Eccentricity: $e

Question

A hyperbola whose transverse axis is along the major axis of the conic, and has vertices at the foci of this conic. If the eccentricity of the hyperbola is then which of the following points does NOT lie on it?

Accepted Answer

This problem involves understanding the properties of both ellipses and hyperbolas, specifically their standard forms, how to determine their axes, foci, vertices, and eccentricity. The key is to carefully extract information from the given ellipse and use it to construct the equation of the hyperbola.

Key Concepts and Formulas:

Standard Equation of an Ellipse centered at the origin:
- If the major axis is along the x-axis ( $a > b$ $a > b$ ): $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$
  - Vertices: $(\pm a, 0)$
  - Foci: $(\pm ae, 0)$
  - Eccentricity: $e

Answer

(0, 2)

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