Question

Locus represented by $x=a(\cosh \theta +\sinh \theta )$ ,$y=b(\cosh \theta -\sinh \theta )$ is

• A. Hyperbola
• B. Parabola
• C. Ellipse
• D. Straight line

Answer 1

Answer: (A)

Hyperbola

The equation for $\cosh \theta$ is

$\cosh \theta =\frac{e^{\theta }+e^{-\theta }}{2}$ ...(i)

Similarly $\sinh \theta =\frac{e^{\theta }-e^{-\theta }}{2}$ ...(ii)

Adding both gives us

$\cosh \theta +\sinh \theta =e^{\theta }$ And

$\cosh \theta -\sinh \theta =e^{-\theta }$

Hence, $x=a(\cosh \theta +\sinh \theta )$

$\Rightarrow x=a{e}^{\theta }$

Similarly $y=b(\cosh \theta -\sinh \theta )$

$\Rightarrow y=b{e}^{-\theta }$

Hence, $x=a{e}^{\theta }$ and $y=b{e}^{-\theta }$

Multiplying both gives us

$xy=ab(e^{\theta }.e^{-\theta })$

$xy=ab$

This represents a hyperbola.