If $|Z - 2| = 2|Z - 1|$, then the value of $\frac{|Re(Z)|}{|a|}$ is (where $Z$ is a complex number and $Re(Z)$ represents the real part of $Z$)
Step-by-Step Solution
Key Concept: Complex number loci can be converted to geometric equations by separating real and imaginary parts.
Let $Z = x + iy$. From $|z - 2| + |y| = 2|(z-1) + iy|$, we have $(x-2)^2 + y^2 = 4[(z-1)^2 + y^2]$. Expanding: $(x-2)^2 + y^2 = 4(x-1)^2 + 4y^2$, which simplifies to $3(x^2 + y^2) - 4x = 0$. This is equivalent to $3(x - \frac{2}{3})^2 + 3y^2 = \frac{4}{3}$. The eccentricity is found from the conic properties: $e = \frac{\text{distance to focus}}{\text{semi-major axis}} = 0.75$.
Correct Answer: 0.75