Question

The coordinates of a point are $a\tan (\theta +\alpha )$ and $b\tan (\theta +\beta )$, where $\theta$ is variable, then locus of the point is

• A. hyperbola
• B. rectangular hyperbola
• C. ellipse
• D. None of the above

Answer 1

The correct option is B rectangular hyperbola

$x=a\tan (\theta +\alpha )$
$\Rightarrow \theta +\alpha ={\tan }^{-1}\frac{x}{a}...\left(1\right)$
$y=b\tan (\theta +\beta )$
$\Rightarrow \theta +\beta ={\tan }^{-1}\frac{y}{b}...\left(2\right)$
Substract equation $\left(2\right)$ from $\left(1\right)$
$\alpha -\beta ={\tan }^{-1}\frac{x}{a}-{\tan }^{-1}\frac{y}{b}$
$\Rightarrow {\tan }^{-1}\frac{\frac{x}{a}-\frac{y}{b}}{1+\frac{x}{a}\cdot \frac{y}{b}}=\alpha -\beta$

$\Rightarrow xy+ab=(bx-ay)\cot (\alpha -\beta )$

Hence, it is the equation of rectangular hyperbola