Question

A rod of length 12 cm 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 the x-axis.

Answer 1

Let AB be the rod making an angle $θ$ with OX and 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 $PR⊥OX.$

In $\triangle PBQ, cos θ = \frac{PQ}{PB} = \frac{x}{9}$

In $\triangle PRA, Sin θ =\frac{ PR}{PA} =\frac{ y}{3}$

Since, $sin^2 θ +cos^2 θ = 1,$

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

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

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