Let the eccentricity of the ellipse be b times the eccentricity of the hyperbola , where a is the mi

Q: JEE Main 2024 Conic Sections: Let the eccentricity of the ellipse be b times the eccentricity of the hyperbola , where a is the mi...

The correct answer is A. This problem is a comprehensive test of concepts from conic sections (ellipse and hyperbola) and differential calculus (finding the minimum distance between curves). We will systematically break down the problem into logical steps, explaining each part in detail. **1. Key Concepts and Formulas** Before diving into the solution, let's recall the essential definitions and formulas: * **Standard Form of Ellipse:** For an ellipse : * If , the major axis is along the x-axis, and eccentricity . * If $

Question

Let the eccentricity of the ellipse be b times the eccentricity of the hyperbola , where a is the minimum distance between the curves y = e x and y = log e x. Then is equal to :

Accepted Answer

This problem is a comprehensive test of concepts from conic sections (ellipse and hyperbola) and differential calculus (finding the minimum distance between curves). We will systematically break down the problem into logical steps, explaining each part in detail.

1. Key Concepts and Formulas

Before diving into the solution, let's recall the essential definitions and formulas:

Standard Form of Ellipse: For an ellipse $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ $\frac{x ^{2}}{A ^{2}} + \frac{y ^{2}}{B ^{2}} = 1$ :
- If $A^2 > B^2$ , the major axis is along the x-axis, and eccentricity $e = \sqrt{1 - \frac{B^2}{A^2}}$ .
- If $

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