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.