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For the ellipse x²/16 + y²/b² = 1 with 0 < b < 4, the latus rectum subtends an angle 2*theta at the farthest vertex such that cosec(theta) = sqrt(5). Which of the following is correct?

1. area of triangle formed by LR and nearest vertex is 6 sq. units
2. e = 1/2
3. no such ellipse exist
4. b = 2*sqrt(3)

Correct answer: area of triangle formed by LR and nearest vertex is 6 sq. units

Solution

From cosec(theta)=sqrt(5), tan(theta)=1/2; combined with tan(theta) = (b²/a)/(a(1+e)) and a=4, solving gives e=1/2, b²=12 (b=2sqrt3); the triangle formed by the LR and the nearest vertex has area (1/2)*(2*b²/a)*(a-ae) = 6.

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