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Question

If the point z1 is the reflection of a point z2 through the line ¯bz+b¯z=c, where b0, then ¯bz1+b¯z2 is

A
Proportional to |z1|2|z2|2
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B
Equal to c
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C
Inversely proportional to c
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D
None of these
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Solution

The correct options are
C Equal to c
D Proportional to |z1|2|z2|2
The given line is the perpendicular bisector of the line joining the 2 given points say A (z1) and B (z2).
Let R (z) be the intersection point of the 2 lines.
Thus, AR = BR => |zz1|=|zz2|=>|zz1|2=|zz2|2=>z(¯z1¯z2)+¯z(z1z2)=|z1|2|z2|2
This is identical to ¯bz+¯zb=c.
Hence, b=z1z2 and c=|z1|2|z2|2
Thus, ¯bz1+b¯z2=(¯z1¯z2)z1+(z1z2)¯z2=|z1|2|z2|2=c
Hence, (a), (b) are correct.

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