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Question

# If ∣∣∣z−z1z−z2∣∣∣=3, where z1 and z2 are fixed complex numbers and z is a variable complex number, then 'z' lies on a

A
Circle with centre as 9z1z28
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B
Circle with z2 as its interior point
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C
Circle with centre as 9z2z18
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D
Circle with z2 as its exterior point
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Solution

## The correct option is C Circle with centre as 9z2−z18Let P1,P2 be two points on the line joining the points A(α),B(β) which divides A and B in the ration 3:1 internally and externally Now internal and external bisectors of ∠APB will meet the line joining points A and B at P1 and P2, respectively. Since, AP1:P1B≡PA:PB≡3:1 (internal division) and AP2:P2B≡PA:PB≡3:1 (external division) ⇒∠P1PP2=π2 Thus locus of 'P' is a circle having P1P2 as its diameter. Now P1=z1+3z24,P2=3z2−z12 Hence centre of circle will be =P1+P22 =9z2−z18

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