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Question

# If the lines a¯¯¯z+¯¯¯az+b=0 and c¯¯¯z+¯¯cz+d=0 are mutually perpendicular, where a and c are non zero complex numbers, while b and d are real numbers, then

A
a¯¯¯a+c¯¯c=0
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B
a¯¯c is purely imaginary
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C
arg(ac)=±π2
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D
a¯¯¯a=c¯¯c
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Solution

## The correct option is C arg(ac)=±π2Let a=a1+ia2 and c=c1+ic2. Then, a1a2×c1c2=−1 ⇒a1c1+a2c2=0 ⇒(a+¯¯¯a2)(c+¯¯c2)+(a−¯¯¯a2i)(c−¯¯c2i)=0 ⇒a¯¯c+¯¯¯ac=0 ⋯(1) ∴a¯¯c is purely imaginary. From (1), we have ac=−¯¯¯a¯¯c ⇒ac+¯¯¯a¯¯c=0 ⇒ac is purely imaginary. ∴arg(ac)=±π2

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