Question

The co-ordinate of a moving point $P$ is $\left(3\right(\cot \theta +tan\theta ),4(\cot \theta -\tan \theta \left)\right)$. Then find the locus of the point $P$.

Answer 1

Let $(x,y)$ be the co-ordinate of the point.

Given, the co-ordinate of a moving point $P$ is $\left(3\right(\cot \theta +\tan \theta ),4(\cot \theta -\tan \theta \left)\right)$.

Then $x=3(\cot \theta +\tan \theta )$ and $y=4(\cot \theta -\tan \theta )$

or, $\frac{x}{3}=\cot \theta +\tan \theta$.....(1) and $\frac{y}{4}=\cot \theta -\tan \theta$......(2).

Now, from (1) and (2) we get,

${\left(\frac{x}{3}\right)}^2$$-{\left(\frac{y}{4}\right)}^2=4.\cot \theta .\tan \theta$ [ Since $(a+b{)}^2-(a-b{)}^2=4ab$]

or, ${\left(\frac{x}{3}\right)}^2$$-{\left(\frac{y}{4}\right)}^2=4$.

This is the locus of the point $P$.