# 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.

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 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 BPQ,

cos =

And in the triangle PAR,

sin =

We know, sin2 + cos2 = 1

This is the locus of the point P.

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