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

Step 1: Interpret the first modulus equation.
\[ \left| \frac{z - 6i}{z - 2i} \right| = 1 \Rightarrow |z - 6i| = |z - 2i| \] This represents the locus of points equidistant from $6i$ and $2i$, which is the perpendicular bisector of the line joining these points.
Hence, \[ \text{Im}(z) = 4 \] Step 2: Interpret the second modulus equation.
\[ \left| \frac{z - (8 - 2i)}{z + 2i} \right| = \frac{3}{5} \Rightarrow |z - (8 - 2i)| = \frac{3}{5}|z + 2i| \] This represents a circle (Apollonius circle).
Let $z = x + iy$ and substitute $y = 4$: \[ |(x + 4i) - (8 - 2i)| = \frac{3}{5}|(x + 4i) + 2i| \] \[ \sqrt{(x - 8)^2 + 6^2} = \frac{3}{5}\sqrt{x^2 + 6^2} \] Step 3: Solve the equation.
Squaring both sides: \[ (x - 8)^2 + 36 = \frac{9}{25}(x^2 + 36) \] \[ 25(x - 8)^2 + 900 = 9x^2 + 324 \] \[ 16x^2 - 400x + 2176 = 0 \] \[ x = 8, \; 17 \] Step 4: Compute $\sum |z|^2$.
For $z_1 = 8 + 4i$: \[ |z_1|^2 = 8^2 + 4^2 = 80 \] For $z_2 = 17 + 4i$: \[ |z_2|^2 = 17^2 + 4^2 = 305 \] \[ \sum |z|^2 = 80 + 305 = 385 \]