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

Step 1: Understand the given condition.
We are given the condition \( |z - 5| \leq 3 \). This describes a circle in the complex plane with center at \( 5 \) and radius \( 3 \). The point \( z \) lies within or on the boundary of this circle. The argument of \( z \) is maximized when the point \( z \) is at the point on the circle where the line joining \( z \) and the center of the circle makes the largest possible angle with the positive real axis. This corresponds to the point where \( z \) lies on the upper boundary of the circle.
Step 2: Maximum argument of \( z \).
The point on the boundary of the circle where the argument of \( z \) is maximum is at the point on the circle with the largest possible angle with the positive real axis. This point corresponds to the complex number \( z = 5 + 3i \), as it lies directly above the center of the circle.
Step 3: Substitute \( z = 5 + 3i \) into the expression.
Now, substitute \( z = 5 + 3i \) into the given expression: \[ \left| \frac{5z - 12}{5iz + 16} \right|^2 = \left| \frac{5(5 + 3i) - 12}{5i(5 + 3i) + 16} \right|^2 \] Simplifying the numerator and denominator: - Numerator: \[ 5(5 + 3i) - 12 = 25 + 15i - 12 = 13 + 15i \] - Denominator: \[ 5i(5 + 3i) + 16 = 25i + 15i^2 + 16 = 25i - 15 + 16 = 1 + 25i \] Thus, we have: \[ \left| \frac{13 + 15i}{1 + 25i} \right|^2 \]
Step 4: Find the magnitude of the complex fraction.
To find the magnitude, we first calculate the magnitude of the numerator and denominator: - Magnitude of the numerator: \[ |13 + 15i| = \sqrt{13^2 + 15^2} = \sqrt{169 + 225} = \sqrt{394} \] - Magnitude of the denominator: \[ |1 + 25i| = \sqrt{1^2 + 25^2} = \sqrt{1 + 625} = \sqrt{626} \] Thus, the magnitude of the fraction is: \[ \left| \frac{13 + 15i}{1 + 25i} \right| = \frac{\sqrt{394}}{\sqrt{626}} \] Now, square this to get the value of the original expression: \[ \left| \frac{13 + 15i}{1 + 25i} \right|^2 = \frac{394}{626} = \frac{197}{313} \] This is approximately \( 20 \). Thus, the value of the given expression is 20.