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 Caz+¯¯¯¯¯¯az=0 Given equation of line is a¯¯¯z+¯¯¯az+b=0,∀b∈R 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(¯¯¯a−a)+b=0 ⇒z1=−b¯¯¯a−a Let locus of mid- point be z , z=zR+zI2 =−b2[1¯¯¯a+a+1¯¯¯a−a] =¯¯¯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