Question

If $|\frac{z-z_1}{z-z_2}|$ = 3, Where $z_1$ and $z_2$ 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
• B. Circle with 'z₂' as it's interior point
• C. Circle with 'z₁'and 'z₂' as it's interior points
• D. Circle with 'z₁'and 'z₂' as it's exterior points.

Answer 1

The correct option is B

Circle with 'z₂' as it's interior point

Let the internal and external bisectors of $\angle APB$ meet the line joining $A\left(Z_1\right)$ and $B\left(Z_2\right)$ 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 $\angle MPN$ = $\frac{\pi }{2}$ and therefore P(z) lies on a circle having MN as it diameter. Clearly $B\left(z_2\right)$ is its interior point.