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The locus of the mid-points of the chord...

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 (a) `(r)/(2)` (b) `(2r)/(3)` (c) `(r )/(sqrt(2))` (d) `(r )/(sqrt(3))`

A

`(r)/(2)`

B

`(2r)/(3)`

C

`(r )/(sqrt(2))`

D

`(r )/(sqrt(3))`

Text Solution

Verified by Experts

The correct Answer is:
C
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.
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