A rod AB of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on x-axis and end point B lies on y-axis. A point P(x, y) is taken on the rod in such a way that AP = 6 cm. Show that the locus of P is an ellipse.

Let AB be the rod making an angle θ with OX as shown in the figure, and let P(x, y) be a point on it such that AP = 6 cm.

Since AB = 15 cm, we have PB = 9 cm.

From P, draw PQ and PR perpendiculars on the y-axis and x-axis respectively.

From △PBQ, cosθ = x / 9

From △PRA, sinθ = y / 6

Since cos²θ + sin²θ = 1,

(x / 9)² + (y / 6)² = 1

or equivalently, x² / 81 + y² / 36 = 1

Thus, the locus of P is an ellipse.