Let the foci and length of the latus rectum of an ellipse fracx^2a^2 + fracy^2b^2 = 1, a > b be (pm 5, 0) and sqrt50, respectively. Then, the square of the eccentricity of the hyperbola fracx^2b^2 - fracy^2a^2 b^2 = 1 equals

Numerical Answer Type:
Enter a numerical value Answer: 51 to 51

Solution & Explanation

### Core Logic For the ellipse, foci are at (pm 5, 0) implies ae = 5. Latus rectum = frac2b^2a = sqrt50 = 5sqrt2 implies b^2 = frac5sqrt2a2. ### Step 1: Solve for a and b Using b^2 = a^2(1 - e^2): a^2 - (ae)^2 = b^2 implies a^2 - 25 = frac5sqrt2a2 2a^2 - 5sqrt2a - 50 = 0 2a^2 - 10sqrt2a + 5sqrt2a - 50 = 0 2a(a - 5sqrt2) + 5sqrt2(a - 5sqrt2) = 0 a = 5sqrt2 (since a > 0). Now, b^2 = frac5sqrt2(5sqrt2)2 = 25 implies b = 5. ### Step 2: Hyperbola Eccentricity The hyperbola is fracx^2b^2 - fracy^2a^2b^2 = 1. Here, semi-major axis A = b and semi-minor axis B = ab. Using eccentricity formula for hyperbola e_H^2 = 1 + fracB^2A^2: e_H^2 = 1 + fraca^2 b^2b^2 = 1 + a^2 Since a = 5sqrt2, a^2 = 50. e_H^2 = 1 + 50 = 51 ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Conic Sections

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