Let the eccentricity of an ellipse x 2 a 2 + y 2 b 2 = 1 , a > b , be 1 4 . If this ellipse passes through the point - 4 2 5 , 3 , then a 2 + b 2 is equal to

Step-by-step solution

Given, x 2 a 2 + y 2 b 2 = 1    a > b Now using eccentricity formula, e 2 = 1 - b 2 a 2 We get, 1 16 = 1 - b 2 a 2 ⇒ b 2 a 2 = 1 - 1 16 = 15 16 ⇒ b 2 = 15 16 a 2 Now again x 2 a 2 + y 2 b 2 = 1 is passing through - 4 2 5 , 3 on satisfying the point we get, ⇒ 16 × 2 5 a 2 + 9 b 2 = 1 ⇒ 32 5 a 2 + 9 b 2 = 1 Now putting the value b 2 = 15 16 a 2 in above equation we get, 32 5 a 2 + 9 15 16 a 2 = 1 80 5 a 2 = 1 16 = a 2 So, b 2 = 15 and a 2 + b 2 = 15 + 16 = 31