The vertices of a hyperbola H are ( 6, 0) and its eccentricity is . Let N be the normal to H at a po

Q: JEE Main 2023 Conic Sections: The vertices of a hyperbola H are ( 6, 0) and its eccentricity is . Let N be the normal to H at a po...

The correct answer is 2. This problem combines concepts of hyperbolas, including their standard equation, eccentricity, properties of normals, and coordinate geometry principles such as slope and distance. We will systematically break down the problem into clear, manageable steps. --- **1. Determine the Equation of the Hyperbola H** * **Key Concept:** The standard equation of a hyperbola with its transverse axis along the x-axis (vertices on the x-axis) is given by . The vertices are at . The eccentricity of a hyperbola

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The vertices of a hyperbola H are ( 6, 0) and its eccentricity is . Let N be the normal to H at a point in the first quadrant and parallel to the line . If d is the length of the line segment of N between H and the y-axis then d is equal to .

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This problem combines concepts of hyperbolas, including their standard equation, eccentricity, properties of normals, and coordinate geometry principles such as slope and distance. We will systematically break down the problem into clear, manageable steps.

1. Determine the Equation of the Hyperbola H

Key Concept: The standard equation of a hyperbola with its transverse axis along the x-axis (vertices on the x-axis) is given by $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ . The vertices are at $(\pm a, 0)$ . The eccentricity $e$ of a hyperbola is related to $a$ and $b$ by the formula $$e^2 = 1 + \

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