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 the x -axis.
Given, length of rod with ends touching coordinate axes is 12 cm .
Let, AB be the rod making an angle θ with OX and P ( x , y ) be the point on it such that AP = 3 cm
As AB = 12 . Then,
PB = AB − AP = 12 − 3 = 9
From P , draw PQ ⊥ QY and PR ⊥ QX .
In Δ PBQ ,
cos θ = PQ PB = x 9
In Δ PRA ,
sin θ = PR PA = y 3
We know that sin 2 θ + cos 2 θ = 1 . So,
( y 3 ) 2 + ( x 9 ) 2 = 1 x 2 81 + y 2 9 = 1
Thus, the equation of the locus of point P is x 2 81 + y 2 9 = 1 .
Given, length of rod with ends touching coordinate axes is
Let, AB be the rod making an angle
As
From
We know that
Thus, the equation of the locus of point P is
