Question

If \( z_1 = 2 + 5i \), \( z_2 = -1 + 4i \) and \( z_3 = i \), then
\[ \frac{|z_1 - z_3|}{|z_3 - z_2|} = ? \]

• A. \( \sqrt{2} \)
• B. \( 2\sqrt{2} \)
• C. \( 5\sqrt{2} \)
• D. \( 4\sqrt{2} \)

Hint:

When working with complex numbers, first compute the differences, then apply the modulus formula \( |z| = \sqrt{a^2 + b^2} \), where \( z = a + bi \), and simplify the expression.

Answer 1

The Correct Option is A

We are given the complex numbers: \[ z_1 = 2 + 5i, \quad z_2 = -1 + 4i, \quad z_3 = i \] We need to evaluate the expression: \[ \frac{|z_1 - z_3|}{|z_3 - z_2|} \] Step 1: First, compute \( |z_1 - z_3| \): \[ z_1 - z_3 = (2 + 5i) - i = 2 + 4i \] \[ |z_1 - z_3| = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \] Step 2: Next, compute \( |z_3 - z_2| \): \[ z_3 - z_2 = i - (-1 + 4i) = 1 - 3i \] \[ |z_3 - z_2| = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \] Step 3: Now, substitute these values into the expression: \[ \frac{|z_1 - z_3|}{|z_3 - z_2|} = \frac{2\sqrt{5}}{\sqrt{10}} \] Step 4: Simplify the expression: \[ \frac{2\sqrt{5}}{\sqrt{10}} = \frac{2\sqrt{5}}{\sqrt{2 \times 5}} = \frac{2\sqrt{5}}{\sqrt{5}\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \] Thus, the correct answer is \( \sqrt{2} \).