Question

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

Answer 1

Let AB be the rod making an angle θ with OX and let P (x, y) be the point on it such that AP = 3 cm.
Then, PB = AB – AP = (12 – 3) cm = 9 cm [∵ AB = 12 cm]
From P, draw PQ⊥OY and PR⊥OX.

$$\begin{gathered} \mathrm{In}△PBQ,\mathrm{we}\mathrm{have}: \\ \cos \theta =\frac{PQ}{PB}=\frac{x}{9} \\ \mathrm{In}△PRA,\mathrm{we}\mathrm{have}: \\ \sin \theta =\frac{PR}{PA}=\frac{y}{3} \\ \mathrm{Since}{\sin }^2\theta +{\cos }^2\theta =1,\mathrm{we}\mathrm{have}: \\ {\left(\frac{y}{3}\right)}^2+{\left(\frac{x}{9}\right)}^2=1 \\ \Rightarrow \frac{x^2}{81}+\frac{y^2}{9}=1 \\ \mathrm{Thus},\mathrm{the}\mathrm{locus}\mathrm{of}\mathrm{a}\mathrm{point}P\mathit{}\mathrm{on}\mathrm{the}\mathrm{rod}\mathrm{is}\frac{x^2}{81}+\frac{y^2}{9}=1. \end{gathered}$$