Angle made with the x-axis by a straight line drawn through (1, 2) so that it intersects `x+y=4` at a distance `(sqrt(6))/3` from (1, 2) is `105^0` (b) `75^0` (c) `60^0` (d) `15^0`

Options

A. `pi//6 and pi//3`
B. `pi//8 and 3pi//8`
C. `pi//12 and 5pi//12`
D. none of these

Answer

C

Explanation

Suppose a line passing through (1,2) makes an angle `theta` with x-axis . Then, its equation is
`(x-1)/(cos theta)=(y-2)/(sin theta)`
The coordinates of a point on this line at a distance `(sqrt(16))/(3)=sqrt((2)/(3))` from (1,2) are given by
`(x-1)/(costheta)=(y-2)/(sin theta)=sqrt((2)/(3))` or, `(1+sqrt((2)/(3)) costheta, 2+sqrt((2)/(3)) sin theta)`
This point lies on `x+y=4`
`therefore 1+sqrt((2)/(3)) costheta + 2+ sqrt((2)/(3)) sin theta=4`
`implies cos theta + sin theta=sqrt((3)/(2))`
`implies sin (theta+(pi)/(4))=(sqrt(3))/(2)implies theta=(pi)/(12) and (5pi)/(12)`