Let an ellipse with centre 1 , 0 and latus rectum of length 1 2 have its major axis along x-axis. If its minor axis subtends an angle 60 ∘ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to _______.

Step-by-step solution

Given, An ellipse with centre 1 ,   0 and latus rectum of length 1 2 have its major axis along x-axis And its minor axis subtends an angle 60 ∘ at the foci, So, on plotting the diagram we get, Now from diagram we get, 2 b 2 a = 1 2 ⇒ b 2 = a 4   . . . 1 And   tan 30 ∘ = b a e ⇒ 1 3 2 = b a e 2 ⇒ 1 3 = b 2 a 2 - b 2 ⇒ a 2 - b 2 = 3 b 2 ⇒ b 2 = a 2 4     . . . . . . . 2 Now from equation 1   &   2 we get, ⇒ a = 1 ,   b 2 = 1 4