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

Step-by-step solution

∵   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 Therefore, the length of latus rectum = 2 b 2 a = 2 × 27 6 = 9