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))`

Options

A. `x^(2)+y^(2)=1`
B. `x^(2)+y^(2)=(27)/(4)`
C. `x^(2)+y6(2)=(9)/(4)`
D. `x^(2)+y^(2)=(3)/(2)`

Answer

C

Explanation

Let P(h,k) be the mid-point of the chord subtending angle `(2pi)/(3)` at the centre.
In `DeltaOPA`, we have
`cos (pi)/(3)=(OP)/(OA)`
`rArr 2OP=OA`
`rArr 4 OP^(2)=OA^(2)`
`rArr 4(h^(2)+k^(2))=9`
`:. `Locus of (h,k) is `x^(2)+y^(2)=(9)/(4)`.