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.

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