Let be an ellipse. Ellipses 's are constructed such that their centres and eccentricities are same a

Q: JEE Main 2023 Conic Sections: Let be an ellipse. Ellipses 's are constructed such that their centres and eccentricities are same a...

The correct answer is 1. This problem involves a sequence of ellipses, each defined by specific geometric properties related to the previous one. We will utilize the standard formulas for ellipses, including eccentricity and area, and then identify a geometric progression to find the sum of their areas. --- **1. Key Concepts and Formulas for Ellipses** For an ellipse centered at the origin, with its major axis along the x-axis: * **Standard Equation:** , where is the semi-major axis and is the semi-minor axis, with . *

Question

Let be an ellipse. Ellipses 's are constructed such that their centres and eccentricities are same as that of , and the length of minor axis of is the length of major axis of . If is the area of the ellipse , then , is equal to .

Accepted Answer

This problem involves a sequence of ellipses, each defined by specific geometric properties related to the previous one. We will utilize the standard formulas for ellipses, including eccentricity and area, and then identify a geometric progression to find the sum of their areas.

1. Key Concepts and Formulas for Ellipses

For an ellipse centered at the origin, with its major axis along the x-axis:

Standard Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , where $a$ is the semi-major axis and $b$ is the semi-minor axis, with $a > b$ .
Length of Major Axis: $2a$
**Length

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