If a point `z_(1)` is the reflection of a point `z_(2)` through the line `b barz+barbz=c, b in 0`, in the Argand plane, then `b barz_(2)+ barb z_(1)=`

Options

A. 4c
B. 2c
C. c
D. none of these

Answer

C

Explanation

If `P(z_(1))` is the reflection of `Q(z_(1))` through the line `bbarz+barbz=c` in the Argand plane. Then, `R(z_(1)+z_(2))/2` lies on the line.

`therefore b(barz_(1)+barz_(2))/(2) + barb(barz_(1)+barz_(2))/(2)=c`
`rArr bbarz_(1)+barbz_(1)+bbarz_(2)+barbz_(2)=2c`............(i)
Since, PQ is perpendicular to the line `bbarz+barbz=c`. Therefore,
Slope of PQ+ slope of the line =0
`(z_(2)-z_(1))/(z_(2)-barz_(1))+(-b/barb)=0`
`rArr barb(z_(2)-z_(1))-b(barz_(2)-barz_(1))=0`
`rArr barbz_(2)-barbz_(1)-bbarz_(2)+bbarz_(1)=0` .................(ii)
Adding (i) and (ii), we get
`2(bbarz_(1)+barbz_(2))=2c rArr bbarz_(1)+barbz_(2)=c`.