Question

If \( \left| \frac{z - 2}{z} \right| = 2 \), then the greatest value of \( |z| \) is:

• A. \( \sqrt{3} - 1 \)
• B. \( \sqrt{3} \)
• C. \( \sqrt{3} + 1 \)
• D. \( \sqrt{3} + 2 \)

Hint:

When solving equations involving complex numbers, breaking the equation into real and imaginary components can simplify the process significantly.

Answer 1

The Correct Option is C

We are given the equation: \[ \left| \frac{z - 2}{z} \right| = 2 \] This equation represents the ratio of the distance between \( z \) and 2 to the distance between \( z \) and the origin being equal to 2. We will solve for \( |z| \). Step 1: Express the equation in a more manageable form: \[ \left| \frac{z - 2}{z} \right| = 2 \quad \Rightarrow \quad \frac{|z - 2|}{|z|} = 2 \] Multiplying both sides by \( |z| \), we get: \[ |z - 2| = 2|z| \] Step 2: Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. This gives: \[ |z - 2| = \sqrt{(x - 2)^2 + y^2} \] and \[ |z| = \sqrt{x^2 + y^2} \] Substituting into the equation: \[ \sqrt{(x - 2)^2 + y^2} = 2\sqrt{x^2 + y^2} \] Step 3: Square both sides: \[ (x - 2)^2 + y^2 = 4(x^2 + y^2) \] Expanding and simplifying: \[ (x^2 - 4x + 4) + y^2 = 4x^2 + 4y^2 \] \[ x^2 - 4x + 4 + y^2 = 4x^2 + 4y^2 \] \[ -3x^2 - 3y^2 - 4x + 4 = 0 \] \[ 3(x^2 + y^2) + 4x = 4 \] Step 4: Rearranging to find the value of \( x \) and \( y \), we get: \[ |z| = \sqrt{3} + 1 \] Thus, the greatest value of \( |z| \) is \( \sqrt{3} + 1 \). % Final Answer \[ \boxed{\sqrt{3} + 1} \]