Question

A rod of length 15 cm moves with its ends always touching the coordinate axes. Find the equation of the locus of a point P on the rod, which is at a distance of 3 cm from the end in contact with the x-axis.

Answer 1

Given: A rod of length 15 cm moves with its ends always touching the coordinate axes. A point P on the rod, which is at a distance of 3 cm from the end in contact with the x-axis

Need to find: Find the equation of the locus of a point P

Here AB is the rod making an angle θ with the x-axis. 

Here AP = 3. 

PB = AB – AP = 12 – 3 = 9 cm 

Here, PQ is the perpendicular drawn from the x-axis and RP is the perpendicular drawn from y-axis. 

Let, the coordinates of the point P is (x, y). 

Now, in the triangle ΔBPQ,

cos θ = x/PB = x/9

And in the triangle ◬PAR

sin θ = y/AP = y/3

We know. sin²θ + cos²θ = 1

⇒ x²/81 +y²/9 = 1

This is the locus of the point P.