"Let $H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$, a 0, b 0, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4(2\sqrt 2 + \sqrt {14} )$. If the eccentricity H is ${{\sqrt {11} } \over 2}$, then the value of a2 + b2 is equal to __________."

Question

Accepted Answer

"

$2a + 2b = 4\left( {2\sqrt 2 + \sqrt {14} } \right)$ ...... (1)

$1 + {{{b^2}} \over {{a^2}}} = {{11} \over {14}}$ ....... (2)

$\Rightarrow {{{b^2}} \over {{a^2}}} = {7 \over 4}$ ....... (3)

and $a + b = 4\sqrt 2 + 2\sqrt {14}$ ...... (4)

By (3) and (4)

$\Rightarrow a + {{\sqrt 7 } \over 2}a = 4\sqrt 2 + 2\sqrt {14}$

$$ \Rightarrow {{a\left( {2 + \sqrt 7 } \right)} \over 2} = 2\sqrt 2 \left( {2 + \sqrt 7 } \right)$$

$\Rightarrow a = 4\sqrt 2 \Rightarrow {a^2} = 32$ and ${b^2} = 56$

$\Rightarrow {a^2} + {b^2} = 32 + 56 = 88$

"

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

Solution

About this question

Drill 25 more like these. Every day. Free.

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

Solution

About this question

More Conic Sections questions

Drill 25 more like these. Every day. Free.

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