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 A Hyperbola

Given that,
$x=a\tan (\theta +\alpha )$
and $y=b\tan (\theta +\beta )$
or ${\tan }^{-1}\frac{x}{a}=\theta +\alpha ....\left(i\right)$
and ${\tan }^{-1}\frac{y}{b}=\theta +\beta ....\left(ii\right)$
To get the required locus, we have to eliminate $\theta$ from Eqs. (i) and (ii).
On subtracting Eq. (ii) from Eq. (i), we get
${\tan }^{-1}\left(\frac{x}{a}\right)-{\tan }^{-1}\left(\frac{y}{b}\right)=\alpha -\beta$
$\Rightarrow {\tan }^{-1}\left\{\frac{(\frac{x}{a}-\frac{y}{b})}{1+\frac{x}{a}\cdot \frac{y}{b}}\right\}=\alpha -\beta$
$\Rightarrow \frac{\frac{x}{a}-\frac{y}{b}}{1+\frac{x}{a}\cdot \frac{y}{b}}=\tan (\alpha -\beta )$
Simplifying, we get the required locus as
$xy+ab=(bx-ay)\cot (\alpha -\beta )$
which is a hyperbola.