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Question

Let P(z) be a point in complex plane satisfying z¯¯¯z+(45i)¯¯¯z+(4+5i)z=40. If a=max|z+23i| and b=min|z+23i|, then

A
a+b=18
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B
a+b=9
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C
ab=42
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D
ab=73
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Solution

The correct option is D ab=73
General equation of circle in complex form is z¯¯¯z+a¯¯¯z+¯¯¯az+b=0,
where centre is a and radius is a¯¯¯ab
So, centre of circle is C(4,5)
and radius, r=(4)2+(5)2+40=9

Distance of centre C(4,5) from P(2,3) is 22<r=9
So, 2+3i lies inside the circle.


a=max|z(2+3i)|=r+CP=9+22
and b=min|z(2+3i)|=rCP=922

Hence, a+b=18
ab=42
and ab=(9+22)(922)=818=73

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