The length of the minor axis (along y -axis) of an ellipse in the standard form is 4 3 . If this ellipse touches the line x + 6 y = 8 then its eccentricity is:

Step-by-step solution

Let equation of ellipse be x 2 a 2 + y 2 b 2 = 1 . Here, 2 b = 4 3 ⇒ b = 2 3 Equation of tangents ≡ y = m x ± a 2 m 2 + b 2 Comparing with ≡ y = - x 6 + 4 3 m = - 1 6 and a 2 m 2 + b 2 = 16 9 ⇒ a 2 36 + 4 3 = 16 9 ⇒ a 2 36 = 16 9 - 4 3 = 4 9 ⇒ a 2 = 16 Now, e = 1 - b 2 a 2 ⇒ e = 1 - 4 3 × 16 = 11 12 ⇒ e = 1 2 11 3