Question

The curve represented by $x=a(\sin h\theta +\cos h\theta )$,$y=b(-\sin h\theta +\cos h\theta )$ is

• A. a hyperbola
• B. a parabola
• C. an ellipse
• D. a circle

Answer 1

The correct option is A

a hyperbola

Explanation for correct option:

Identifying the type of curve,

Given that the equation of the curve is

$x=a(\sin h\theta +\cos h\theta )\to \left(i\right)$

$y=b(-\sin h\theta +\cos h\theta )\to \left(ii\right)$

Now multiplying the equation $\left(i\right)$and $\left(ii\right)$ we get,

$\begin{array}{l}x\times y=a(\sinh \theta +\cosh \theta )\times b(-\sinh \theta +\cosh \theta )\\ xy=ab\left({\cosh }^2\theta -{\sinh }^2\theta \right)\\ \Rightarrow \frac{xy}{ab}={\cosh }^2\theta -{\sinh }^2\theta \end{array}$

We know that

$\sinh \theta =\frac{e^x-e^{-x}}{2}$

$\cosh \theta =\frac{e^x+e^{-x}}{2}$

Therefore, ${\cosh }^2\theta -{\sinh }^2\theta =1$

Hence, $\begin{array}{l}\frac{xy}{ab}=1\end{array}$(Hyperbola identity)

We can say

$xy=ab$ [the obtained equation is in the form of $xy=\text{constant}$]

This is the form of a rectangular hyperbola.

Hence, option (A) is the correct answer.