`theta_1 and theta_2` are the inclination of lines `L_1 and L_2` with the x-axis. If `L_1 and L_2` pass through `P(x,y)`, then the equation of one of the angle bisector of these lines is

Options

A. `(x-x_(1))/("cos"((theta_(1)+theta_(2))/(2))) = (y-y_(1))/("sin"((theta_(1)+theta_(2))/(2)))`
B. `(x-x_(1))/("-sin"((theta_(1)-theta_(2))/(2))) = (y-y_(1))/("cos"((theta_(1)-theta_(2))/(2)))`
C. `(x-x_(1))/("sin"((theta_(1)+theta_(2))/(2))) = (y-y_(1))/("cos"((theta_(1)+theta_(2))/(2)))`
D. `(x-x_(1))/("-sin"((theta_(1)+theta_(2))/(2))) = (y-y_(1))/("cos"((theta_(1)+theta_(2))/(2)))`

Answer

A, D

Explanation

From the figure, angle bisectors `B_(1) " and " B_(2)` have inclinations of
`(theta_(1) + theta_(2))/(2) " and " ((pi)/(2) + (theta_(1) + theta_(2))/(2))` with the x-axis.
Therefore, equations of angle bisectors are
`(x-x_(1))/("cos"((theta_(1) + theta_(2))/(2))) = (y-y_(1))/("sin"((theta_(1) + theta_(2))/(2)))`
`"and " (x-x_(1))/("cos"((pi)/(2)+(theta_(1) + theta_(2))/(2))) = (y-y_(1))/("sin"((pi)/(2)+(theta_(1) + theta_(2))/(2)))`
`"or " (x-x_(1))/("-sin"((theta_(1) + theta_(2))/(2))) = (y-y_(1))/("cos"((theta_(1) + theta_(2))/(2)))`