# 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.

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