Let e 1 and e 2 be the eccentricities of the ellipse and the hyperbola , respectively. If b

Q: JEE Main 2023 Conic Sections: Let e 1 and e 2 be the eccentricities of the ellipse and the hyperbola , respectively. If b < 5 and ...

The correct answer is A. This problem is an excellent test of your understanding of conic sections, specifically ellipses and hyperbolas, their eccentricities, and the locations of their foci. We will systematically break down the problem, calculate the necessary parameters, and finally determine the eccentricity of the new ellipse. --- **1. Understanding the Properties of the First Ellipse** The first ellipse is given by the equation . Let its semi-major axis be and semi-minor axis be . From the equation, we have and (

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Let e 1 and e 2 be the eccentricities of the ellipse and the hyperbola , respectively. If b < 5 and e 1 e 2 = 1 , then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :

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This problem is an excellent test of your understanding of conic sections, specifically ellipses and hyperbolas, their eccentricities, and the locations of their foci. We will systematically break down the problem, calculate the necessary parameters, and finally determine the eccentricity of the new ellipse.

1. Understanding the Properties of the First Ellipse

The first ellipse is given by the equation $\frac{x^2}{b^2} + \frac{y^2}{25} = 1$ . Let its semi-major axis be $A_1$ and semi-minor axis be $B_1$ . From the equation, we have $A_1^2 = 25$ and $B_1^2 = b^2$ (or vice versa).

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