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Question

The locus of mid-point of line segment intercepted between real and imaginary axes by the line a¯¯¯z+¯¯¯az+b=0 where b is a real parameter and a is a fixed complex number with non- zero real and imaginary parts, is

A
az+¯¯¯az=0
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B
a¯¯¯z+¯¯¯az=0
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C
az+¯¯¯¯¯¯az=0
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D
az¯¯¯az=0
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Solution

The correct option is C az+¯¯¯¯¯¯az=0
Given equation of line is
a¯¯¯z+¯¯¯az+b=0, bR
Let PQ be the segment intercepted between the axes. For real intercept zR.
zR=¯¯¯¯¯¯zR
zR(a+¯¯¯a)+b=0
zR=b(a+¯¯¯a)
For imaginary intercept zI,
zI+¯¯¯¯¯zI=0
zI(¯¯¯aa)+b=0
z1=b¯¯¯aa
Let locus of mid- point be z ,
z=zR+zI2
=b2[1¯¯¯a+a+1¯¯¯aa]
=¯¯¯ab(a+¯¯¯a)(a¯¯¯a)
=¯¯¯aba2(¯¯¯a)2
za2(¯¯¯a)2¯¯¯a=b
As b is real parameter,
za2(¯¯¯a)2¯¯¯a=¯¯¯z(¯¯¯a)2(a)2a
az+¯¯¯¯¯¯az=0

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