Question

A rod of length $12$ cm moves with it ends always touching the coordinate axes. Dertermine the equation of the locus of a point $P$ on the rod which is $3$ cm from the end in contact with the $x$-axis

Answer 1

Let $AB$ be the rod making an angle $\theta$ with positive direction of $x$-axis and $P(x,y)$ be the point on it such that $AP=3cm$

Now, $PB=AB-AP=(12-3)cm=9cm$ $(AB=12cm)$

Draw $PQ\perp OY$ and $PR\perp OX$

In $△PBQ$,

$\cos \theta =\frac{PQ}{PB}=\frac{x}{9}$

In $△PRA$,

$\sin \theta =\frac{PR}{PA}=\frac{y}{3}$

Since ${\sin }^2\theta +{\cos }^2\theta =1$

$\Rightarrow {\left(\frac{y}{3}\right)}^2+{\left(\frac{x}{9}\right)}^2=1$

$\Rightarrow \frac{x^2}{81}+\frac{y^2}{9}=1$

Thus the equation of the locus of point $P$ on the rod is $\frac{x^2}{81}+\frac{y^2}{9}=1$