(x – 1) 2 + y 2 = 9

Sol. Method - 1

If the river is still, the man will be at a distance 3 meters from origin O after 1 second. The locus of all the point where man can reach at t = 1 second is a semicircle of radius 3 and centre at O (dotted semicircle shown in figure).

The river flows to right with a speed 1 m/s. Hence there shall be additional shift in position by
1 m/s × 1 sec = 1m towards right. Hence the locus of all points giving possible position after one second will be the dotted semicircle shifted to right by 1 m as shown in figure.

Hence locus all the points where the man can be at t = 1 sec. is a semicircle of radius 3 and centre at O ′ (1 m, 0 m)

∴ Equation of locus of all the points is

(x – 1) 2 + (y – 0) 2 = 3 2

or (x – 1) 2 + y 2 = 9

Method - 2

Let the relative velocity of the man make angle ‘ θ ’ with the x-axis.

Then at time ‘t’ :

x = (3 cos θ + 1) t

and y = 3 sin θ t

⇒ (x – t) 2 + y 2 = (3 cos θ ) 2 t 2 + (3 sin θ ) 2 t 2

(x – t) 2 + y 2 = 9t 2

at t = 1 sec. the required equation is (x – 1) 2 + y 2 = 9.