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NDA 1 2024 Maths, question 65

Topic: Analytical geometry

Q65

What is the eccentricity of the ellipse if the angle between the straight lines joining the foci to an extremity of the minor axis is 90∘90^{\circ} ?

  1. (a) 13\frac{1}{3}
  2. (b) 12\frac{1}{2}
  3. (c) 13\frac{1}{\sqrt{3}}
  4. (d) 12\frac{1}{\sqrt{2}}Correct answer

Solution.

Answer: (d) 12\frac{1}{\sqrt{2}}

  1. Let the foci be (±c, 0) and a minor-axis endpoint be (0, b).
  2. The two vectors from this endpoint to the foci are perpendicular precisely when b² − c² = 0.
  3. Thus b² = c² and a² = b² + c² = 2c².
  4. The eccentricity c/a is 1/√2.