Question

Let $z$ be the complex number satisfying $|z - 5| \le 3$ and having maximum positive principal argument. Then $34 \left| \frac{5z - 12}{5iz + 16} \right|^2$ is equal to :

• A. 12
• B. 16
• C. 26
• D. 20

Hint:

The point of maximum argument for a circle $|z-z_0|=r$ (where $z_0$ is real) satisfies $\sin \theta = r/|z_0|$.

Answer 1

The Correct Option is D

To solve this problem, we need to determine the complex number \( z \) that satisfies the given conditions and compute the expression \( 34 \left| \frac{5z - 12}{5iz + 16} \right|^2 \). 

1. First, consider the condition \( |z - 5| \le 3 \). This describes a closed disk in the complex plane centered at \( 5 \) (the real part) with a radius of \( 3 \). It contains all complex numbers \( z \) that lie within or on the boundary of this disk.
2. For the second part of the condition, we need to find the value of \( z \) with the maximum positive principal argument. The principal argument \( \text{arg}(z) \) is the angle \( \theta \) from the positive real axis to the line segment joining the origin and the point \( z \) in the complex plane. We want the point on the boundary of this disk that has the largest positive angle with the real axis, which will be the topmost point of the disk closest to the imaginary axis.
3. The center of the disk is \( 5 + 0i \), so moving vertically upwards from this point, the topmost boundary of the disk for maximum argument will be \( z = 5 + 3i \).
4. Substitute \( z = 5 + 3i \) into the expression \(\left| \frac{5z - 12}{5iz + 16} \right|\):
   • Calculate the numerator: \[ 5z - 12 = 5(5 + 3i) - 12 = 25 + 15i - 12 = 13 + 15i \]
   • Calculate the denominator: \[ 5iz + 16 = 5i(5 + 3i) + 16 = (25i + 15i^2) + 16 = 25i - 15 + 16 = 1 + 25i \]
5. Now compute the modulus: \[ \left| \frac{13 + 15i}{1 + 25i} \right| = \frac{|13 + 15i|}{|1 + 25i|} \]
   • Calculate \( |13 + 15i| = \sqrt{13^2 + 15^2} = \sqrt{169 + 225} = \sqrt{394} \).
   • Calculate \( |1 + 25i| = \sqrt{1^2 + 25^2} = \sqrt{1 + 625} = \sqrt{626} \).
6. Thus, \[ \left| \frac{13 + 15i}{1 + 25i} \right| = \frac{\sqrt{394}}{\sqrt{626}} \]
7. Finally, compute the required expression: \[ 34 \left| \frac{5z - 12}{5iz + 16} \right|^2 = 34 \left( \frac{\sqrt{394}}{\sqrt{626}} \right)^2 = 34 \frac{394}{626} \] \[ = 34 \times \frac{1}{2} = 17 \times 2 = 20 \]

Thus, the final answer is 20. Therefore, the correct option is 20.