The circle which can be drawn to pass through (1, 0) and (3, 0) and to touch the y-axis intersect at angle `thetadot` Then `costheta` is equal to `1/2` (b) `-1/2` (c) `1/4` (d) `-1/4`

Options

A. `1//2`
B. `-1//2`
C. `1//4`
D. `-1//4`

Answer

B

Explanation

We have, `C_(1)A =C_(2)A=OD=2`

In `DeltaOP_(1)C_(1)`, we have
`OC_(1)^(2)=OP_(1)^(2)+C_(1)P_(1)^(2)`
`rArr 4+3=OP_(1)^(2)+2^(2)rArr OP_(1)=sqrt(3)`
Thus, the coordinates of `C_(1)` and `C_(2)` are `(2, sqrt(3))` and `(2, -sqrt(3))`
Thus, the coordinates of `C_(1) and C_(2)` are `(2, sqrt(3)) and (2, - sqrt(3))` respectively.

`cos theta = (AC_(1)^(2)+AC_(2)^(2)-C_(1)C_(2))/(2AC_(1).AC_(2))=(4+4-12)/(2xx2xx2)=-(1)/(2)`
`rArr theta = (2pi)/(3)`