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 x - axis.

Given that we need to find the locus of the point on the rod whose ends always touching the coordinate axes.

We need to the equation of locus of point P on the rod, which is 3 cm from the end in contact with x - axis.

Let us assume AB be the rod of length 12 cm and P(x,y) be the required point.

From the figure using similar triangles DAP and CBP we get,

⇒

⇒

⇒ q = 3y ..... (1)

⇒

⇒

⇒ ..... - (2)

Now OB = OC + CB

⇒

⇒ .... (3)

⇒ OA = OD + DA

⇒ OA = y + 3y

⇒ OA = 4y .... (4)

Since OAB is a right angled triangle,

⇒ OA² + OB² = AB²

⇒

⇒

⇒

⇒

⇒ x² + 9y² = 81

∴ The equation of the ellipse is x² + 9y² = 81.