Question

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

• A. A hyperbola
• B. A rectangular hyperbola
• C. An ellipse
• D. None of these

Answer 1

The correct option is A A hyperbola

Given that,
$x=atan(\theta +\alpha )andy=btan(\theta +\beta )$
or $ta{n}^{-1}\left(\frac{x}{a}\right)=\theta +\alpha$ …… (i)
and $ta{n}^{-1}\left(\frac{y}{b}\right)=\theta +\beta$ …… (ii)
To get the required locus, we have to eliminate θ from Equations (i) and (ii).
On subtracting Equation (ii) from Equation (i), we get
$ta{n}^{-1}\left(\frac{x}{a}\right)-ta{n}^{-1}\left(\frac{y}{b}\right)=\alpha -\beta$
$\Rightarrow$ $ta{n}^{-1}\left[\frac{\frac{x}{a}-\frac{y}{b}}{1+\frac{x}{a}.\frac{y}{b}}\right]=\alpha -\beta$
$\Rightarrow$ $\frac{\frac{x}{a}-\frac{y}{b}}{1+\frac{x}{a}.\frac{y}{b}}=tan(\alpha -\beta )$
On simplifying, we get the required locus as
xy + ab = (bx - ay) cot (\alpha -\beta )
which is a hyperbola.