Question

# Let ¯¯bz+b¯¯¯z=c,b≠0 be a line in the complex plane. If a point z1 is the reflection of a point z2 through the line, then c is

A
¯¯¯z1b+z2¯¯b2
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B
¯¯¯z1b+z2¯¯b
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C
3(¯¯¯z1b+z2¯¯b)2
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D
2(¯¯¯z1b+z2¯¯b)
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Solution

## The correct option is B ¯¯¯z1b+z2¯¯bGiven that z1 is the reflection of z2 through the line b¯¯¯z+¯¯bz=c Therefore, for any arbitrary point z on the line, we must have |z−z1|=|z−z2| ⇒|z−z1|2=|z−z2|2 ⇒|z|2+|z1|2−z¯¯¯z1−¯¯¯zz1=|z|2+|z2|2−z¯¯¯z2−¯¯¯zz2 ⇒(¯¯¯z2−¯¯¯z1)z+(z2−z1)¯¯¯z=|z2|2−|z1|2 Thus we have b=z2−z1 and c=|z2|2−|z1|2 Now, ¯¯¯z1b+z2¯¯b=¯¯¯z1(z2−z1)+z2(¯¯¯z2−¯¯¯z1)=|z2|2−|z1|2=c

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