Question

Let H: `x^2/a^2 - y^2/b^2` = 1, a > 0, b > 0, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is `4(2sqrt(2) + sqrt(14))`. If the eccentricity H is `sqrt(11)/2`, then the value of a² + 2b² is equal to ______.

Options

• 86

• 87

• 88

• 89

Answer 1

Let H: `x^2/a^2 - y^2/b^2` = 1, a > 0, b > 0, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is `4(2sqrt(2) + sqrt(14))`. If the eccentricity H is `sqrt(11)/2`, then the value of a² + 2b² is equal to 88.

Explanation:

Given: Equation of hyperbola is H: `x^2/a^2 - y^2/b^2` = 1; a > 0, b > 0

And eccentricity of H is e = `sqrt(11)/2`

And sum of length of transverse and conjugate axis is 2a + 2b = `4(2sqrt(2) + sqrt(14))`

As we know, e = `sqrt(1 + b^2/a^2)`

⇒ e² = `1 + b^2/a^2`

⇒ `11/4 = 1 + b^2/a^2`

⇒ b² = `7/4a^2`

⇒  b = `sqrt(7)/2a`

∵ 2a + 2b = `4(2sqrt(2) + sqrt(14))`

⇒ `2a + sqrt(7)a = 4(2sqrt(2) + sqrt(14))`

⇒ `a(2 + sqrt(7)) = 4sqrt(2)(2 + sqrt(7))`

⇒ a = `4sqrt(2)`

⇒ b = `2sqrt(14)`

∴ a² + b² = `(4sqrt(2))^2 + (2sqrt(14))^2`

= 32 + 56

= 88