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  \(\frac{x^2}{81} + \frac{y^2}{9} = 1\)