Question

Let \( z \) be the complex number satisfying \( |z - 5| \leq 3 \) and having maximum possible positive argument. Then the value of \[ \left| \frac{5z - 12}{5iz + 16} \right|^2 \] is equal to:

• A. 20
• B. 17
• C. 7
• D. 21

Hint:

When working with complex numbers, express the magnitudes and arguments of the numbers, and simplify using known properties of complex arithmetic to find the result.

Answer 1

The Correct Option is A

Given:
\[ |z-5|\le 3 \] and \(z\) has the maximum possible positive argument.

Step 1: Geometrical interpretation

The locus \( |z-5|\le 3 \) represents a circle with centre at \( (5,0) \) and radius \(3\).

The point on this circle having maximum positive argument is the uppermost point of the circle.

Hence, \[ z = 5 + 3i \]

Step 2: Substitute \(z = 5 + 3i\)

Evaluate the numerator:

\[ 5z - 12 = 5(5+3i) - 12 = 25 + 15i - 12 = 13 + 15i \]

Evaluate the denominator:

\[ 5iz + 16 = 5i(5+3i) + 16 = 25i + 15i^2 + 16 \]

\[ = 25i - 15 + 16 = 1 + 25i \]

Step 3: Find the modulus squared

\[ \left|\frac{5z-12}{5iz+16}\right|^2 = \frac{|13+15i|^2}{|1+25i|^2} \]

\[ |13+15i|^2 = 13^2 + 15^2 = 169 + 225 = 394 \]

\[ |1+25i|^2 = 1^2 + 25^2 = 1 + 625 = 626 \]

Step 4: Final calculation

\[ \left|\frac{5z-12}{5iz+16}\right|^2 = \frac{394}{626} = \frac{197}{313} \]

Final Answer:

\[ \boxed{\dfrac{197}{313}} \]