Let x 2 a 2 + y 2 b 2 = 1 , a > b be an ellipse with foci F 1 and F 2 Let A O be its semi-minor axis, where O is the centre of the ellipse. The lines A F 1 and A F 2 , when extended, cut the ellipse again at points B and C respectively.

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Let x 2 a 2 + y 2 b 2 = 1 , a &#62; b be an ellipse with foci F 1 and F 2 Let A O be its semi-minor axis, where O is the centre of the ellipse. The lines A F 1 and A F 2 , when extended, cut the ellipse again at points B and C respectively. Suppose that the &#916; A B C is equilateral. Then, the eccentricity of the ellipse is

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Correct answer: D. 1 2

Given, x 2 a 2 + y 2 b 2 = 1 A B C is an equilateral traingles. In Δ A O F 2 , tan 30 ° = O F 2 O A 1 3 = a e b ⇒ b = 3 a e ⇒ b 2 = 3 a 2 e 2 ⇒ b 2 a 2 = 3 e 2 ⇒ 1 - e 2 = 3 e 2 ∵ e = 1 - b 2 a 2 ⇒ 4 e 2 = 1 ⇒ e 2 = 1 4 ⇒ e = 1 2

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