Question

If the point z1 is the reflection of a point z2 through the line ¯bz+b¯z=c, where b≠0, 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|2The 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 => |z−z1|=|z−z2|=>|z−z1|2=|z−z2|2=>z(¯z1−¯z2)+¯z(z1−z2)=|z1|2−|z2|2This is identical to ¯bz+¯zb=c.Hence, b=z1−z2 and c=|z1|2−|z2|2Thus, ¯bz1+b¯z2=(¯z1−¯z2)z1+(z1−z2)¯z2=|z1|2−|z2|2=cHence, (a), (b) are correct.

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