Question

Let $e$ be the eccentricity of a hyperbola $\frac{ x ^2}{ a ^2}-\frac{ y ^2}{ b ^2}=1, f ( e )$ be the eccentricity of a conjugate hyperbola and g(e) be the eccentricity of an ellipse whose major and minor axis coincides with transverse and conjugate axis of hyperbola then the value of $\int\limits_1^{\sqrt{2}} g ( g ( f ( f ( e ))))$ de is equal to

• A. $\frac{1}{2}$
• B. $\sqrt{2}$
• C. 1
• D. $\sqrt{2}+1$

Answer 1

Answer: (A)

$ \because \frac{1}{ e ^2}+\frac{1}{ f ^2( e )}=1 \Rightarrow \frac{1}{ f ^2( e )}=1-\frac{1}{ e ^2}=\frac{ e ^2-1}{ e ^2} $
$\Rightarrow f ( e )=\frac{ e }{\sqrt{ e ^2-1}} $
$\because e ^2=1+\frac{ b ^2}{ a ^2}=\frac{ a ^2+ b ^2}{ a ^2} $
$\text { and } g ( e )=\sqrt{1-\frac{ b ^2}{ a ^2}} \Rightarrow g ^2( e )=\frac{ a ^2- b ^2}{ a ^2} \Rightarrow e ^2+ g ^2( e )=2 \Rightarrow g ( e )=\sqrt{2- e ^2} $
$\therefore f ( f ( e ))=\frac{\sqrt{ e ^2-1}}{\sqrt{\frac{ e ^2}{ e ^2-1}-1}= e } $
$\therefore g ( f ( f ( e )))= g ( e )=\sqrt{2- e ^2} $
$\Rightarrow g ( g ( f ( f ( e ))))=\sqrt{2-\left(2- e ^2\right)}= e $
$\therefore Given \text { integral }=\int\limits_1^{\sqrt{2}} ede =\left[\frac{ e ^2}{2}\right]_1^{\sqrt{2}}=1-\frac{1}{2}=\frac{1}{2}$