in 1/4 th of a circle , the particle will move 2πr/4 distance ( i.e move 90⁰, 1/4 of circle is the quadrant of circle & total dist. or circumference is 2πr) and the displacement will be √2 r ( displacement is the shortest distance covered & as given radius is r so we can find displacement by using r²+r²=√2 r (θ=90^{0,, ,}by using pyth. theorem) now ratio of distance/displacement=( 2 πr/4)/√2r =π/(2√2)

Let us divide a circle in its four quadrants(as u may hav seen in trigonometry). A qaudrant occupies 1/4 th of a circle.

The key idea here is that here displacement will be straight line from one end to another and distance will be the part of circumference covered.

•For a quadrant, distance,

θ = l / r

l = θ × r = (π/2) × r = πr / 2

•diplacement,(hyp. of triangle)

use PT, h = (P^2 + B^2)^1/2  = (r^2 + r^2)^1/2  = r × √2 

Now, divide and u wiil get, π / (2 × √2)... 

2√2:π