A man can swim in still water with a speed of `3 m//s` x and y axis are drawn along and normal to the bank of river flowing to right with a speed of `1 m//s`. The man starts swimming from origin O at t = 0 second. Assume size of man to be negligible. Find the equation of locus of all the possible points where man can reach at t = 1 sec

Options

A. `(x-1)^(2) + y^(2) =3`
B. `(x-1)^(2) + y^(2) -9`
C. `x^(2) + (y-1)^(2) = 3`
D. `x^(2) + (y-1)^(2) = 9`

Answer

B

Explanation

Method -1
If the river is still, the man will be at a distance 3 metres 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 on position by `1 m//s xx 1 sec =1` metre 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 2 and centre at O' (1 m,0m)
`:.` 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 `theta` with the x-axis
Then at time 't'
`x = (3 cos theta + 1)t`
and `y = 3 sin theta t`
`rArr (x-t)^(2) +y^(2) = (3 cos theta)^(2) t^(2) + (3 sin theta)^(2) t^(2)`
`(x - t)^(2) + y^(2) =9t^(2)`
At t= 1 sec. the required equation is
`(x -1)^(2) +y^(2) =9`.