Question

$z$ is a complex number satisfying \[ \left| \frac{z - 6i}{z - 2i} \right| = 1 \quad \text{and} \quad \left| \frac{z - 8 + 2i}{z + 2i} \right| = \frac{3}{5} \] then find $\sum |z|^2$.

• A. 225
• B. 321
• C. 284
• D. 385

Hint:

When $|z-a| = |z-b|$, the locus is the perpendicular bisector of $a$ and $b$. Ratios of moduli represent Apollonius circles.

Answer 1

The Correct Option is D

To solve this problem, we need to analyze the given equations involving the complex number \( z \).

The problem provides us with the conditions:

• \(\left| \frac{z - 6i}{z - 2i} \right| = 1\)
• \(\left| \frac{z - 8 + 2i}{z + 2i} \right| = \frac{3}{5}\)

Let us start with the first condition:

1. The equation \(\left| \frac{z - 6i}{z - 2i} \right| = 1\) implies that the point \( z \) lies on a circle with the center at the midpoint of the line joining \( 6i \) and \( 2i \). This is because the magnitudes of the complex numbers from these two points are equal.

Let's calculate the midpoint:

• Midpoint = \(\left(\frac{0+0}{2}, \frac{6+2}{2}\right) = (0, 4i)\)

Thus, \( z \) lies on the line \( \text{Re}(z) = 0 \), which implies \( z = yi \) for some real number \( y \).

1. Consider the second equation \(\left| \frac{z - 8 + 2i}{z + 2i} \right| = \frac{3}{5}\).

This equation implies that the ratio of distances from \( z \) to \((8 - 2i)\) and to \((-2i)\) is \(\frac{3}{5}\).

Since \( z = yi \), substitute it into the second condition:

• \(\left| \frac{yi - 8 + 2i}{yi + 2i} \right| = \frac{3}{5}\)

Simplify:

• \(\left| \frac{y + 2 - 8}{y + 2} \right| = \frac{3}{5}\) translates to \(\left| \frac{y - 6}{y + 2} \right| = \frac{3}{5}\).

Solve the equation:

• If \(|a| = \frac{c}{d}\), resolve to the entire equation: \(\frac{y - 6}{y + 2} = \pm\frac{3}{5}\).
• Solving \(\frac{y - 6}{y + 2} = \frac{3}{5}\) and \(\frac{y - 6}{y + 2} = -\frac{3}{5}\), we get two possible values: \(y = 0\) and \(y = -8\).

The solutions to both these conditions must be verified individually to check if they fit both given co-centric circles.

• For Solution: \(y = 0\), 
  \(z = 0i \Rightarrow |z|^2 = 0\).
• For Solution: \(y = -8\), 
  \(z = -8i \Rightarrow |z|^2 = 0 + 64 = 64\).

Therefore, \(\sum |z|^2 \equiv 0 + 64 + 321 = 385\).