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Question

# Let P(z) be a point in complex plane satisfying z¯¯¯z+(4−5i)¯¯¯z+(4+5i)z=40. If a=max|z+2−3i| and b=min|z+2−3i|, 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=73General equation of circle in complex form is z¯¯¯z+a¯¯¯z+¯¯¯az+b=0, where centre is −a and radius is √a¯¯¯a−b 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 2√2<r=9 So, −2+3i lies inside the circle. ∴a=max|z−(−2+3i)|=r+CP=9+2√2 and b=min|z−(−2+3i)|=r−CP=9−2√2 Hence, a+b=18 a−b=4√2 and ab=(9+2√2)(9−2√2)=81−8=73

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