Hyperbola 1

Geometry Level 4

Let $x$ be the eccentricity of a hyperbola and $f\left( x \right)$ be the eccentricity of its conjugate hyperbola. Then find $\int _{ 3 }^{ 5 }{ (f\circ f\circ f\circ f\circ f\circ f\circ f\circ f )\left( x \right) } dx$

Clarification: We're composing 8 $f$ 's

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The answer is 8.00.

3 solutions

Agnishom Chattopadhyay
Oct 27, 2015

Note that the conjugate of the conjugate of a hyperbola is the same hyperbola.

Thus, $f(f(x)) = x$ .

Since we have an even number of f's, our integral just simplifies to the integral of x.

$\int^5_3{f^8(x) \; dx} = \int^5_3{x \; dx} = 8$

Rushikesh Joshi
Apr 29, 2015

Use : the conjugate of conjugate of a hyperbola is the same hyperbola . so integral(xdx) ie x^2/2 Put limits . answer is 8.

Archit Tripathi
Oct 26, 2016

If $x$ is the eccentricity of a hyperbola and $f(x)$ is the eccentricity of its conjugate hyperbola, then one can very easily prove that

$x^{-2}$ + $f(x)^{-2}$ = $1$

Which yields $f(x) = \frac{x}{\sqrt{x^{2}-1}}$

And, $f(f(x)) = x$ . Now rest of work everybody has done.

Hyperbola 1

The answer is 8.00.

3 solutions

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