The locus of the mid-points of the chords of the circle of lines radiùs r which subtend an angle `pi/4` at any point on the circumference of the circle is a concentric circle with radius equal to

Options

A. `(r)/(2)`
B. `(2r)/(3)`
C. `(r )/(sqrt(2))`
D. `(r )/(sqrt(3))`

Answer

C

Explanation

Let the equation of the circle be `x^(2) + y^(2) = r^(2)`. The chord which substends and angle `(pi)/(4)` at the circumference will subtend a right angle at the centre. So, chord joining `A(r,0)` and `B(0,r)` subtends a right angle at the centre (0,0). Mid point of AB is `C((r )/(2),(r )/(2))`.
`:. OC =(r )/(sqrt(2))`, which is radius of locus of C.