If the line x - 2 y = 12 is a tangent to the ellipse x 2 a 2 + y 2 b 2 = 1 at the point 3 , - 9 2 , then the length of the latus rectum of the ellipse is

Question

If the line x - 2 y = 12 is a tangent to the ellipse x 2 a 2 + y 2 b 2 = 1 at the point 3 , - 9 2 , then the length of the latus rectum of the ellipse is, 5 units, 12 2 units, 9 units, 8 3 units

MARKS App · Accepted Answer

∵ 3 , - 9 2 lies on x 2 a 2 + y 2 b 2 = 1 ⇒ 9 a 2 + 81 4 b 2 = 1 ……(1) Equation of the tangent at 3 , - 9 2 is 3 x a 2 + - 9 2 y b 2 = 1 & given equation of the tangent is: x - 2 y = 12 ⇒ x 12 + - y 6 = 1 On comparing these equations: a 2 3 = 12 ⇒ a 2 = 36 ⇒ a = 6 2 b 2 9 = 6 ⇒ b 2 = 27 ⇒ b = 3 3 ∴ Length of L . R . = 2 b 2 a = 2 × 27 6 = 9

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