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Question

# If âˆ£âˆ£âˆ£zâˆ’z1zâˆ’z2âˆ£âˆ£âˆ£ = 3, Where z1 and z2 are fixed complex numbers and z is a variable complex number, then 'z' lies on a:

A

Circle with 'z1' as it's interior point

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B

Circle with 'z2' as it's interior point

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C

Circle with 'z1'and 'z2' as it's interior points

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D

Circle with 'z1'and 'z2' as it's exterior points.

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Solution

## The correct option is B Circle with 'z2' as it's interior point Let the internal and external bisectors of ∠APB meet the line joining A(Z1) and B(Z2) at respectively. M and N respectively. AM:BM = AP:BP =3:1 (internal). AN:BN = AP:BP = 3:1(External). Thus,M and N are fixed points and ∠MPN = π2 and therefore P(z) lies on a circle having MN as it diameter. Clearly B(z2) is its interior point.

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