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JEE Main 2023
Conic Sections
Ellipse
Easy

Question

Let E1:x2a2+y2b2=1,a>b{E_1}:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,a > b. Let E 2 be another ellipse such that it touches the end points of major axis of E 1 and the foci of E 2 are the end points of minor axis of E 1 . If E 1 and E 2 have same eccentricities, then its value is :

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Solution

This solution aims to provide a comprehensive and detailed approach to solving the problem, focusing on clarity, educational value, and proper mathematical reasoning.

Key Concepts and Formulas for Ellipses

An ellipse is a set of all points in a plane such that the sum of the distances from two fixed points (foci) is a constant. Its standard equation and properties depend on the orientation of its major axis.

  1. Ellipse with Major Axis along the x-axis (Horizontal Ellipse):
    • Condition: Semi-major axis length aa is greater than semi-minor axis length bb (a>ba > b).
    • Standard Equation: $$\frac{x^2}{a^2} + \frac{y^2}{b
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