Let z be the complex number satisfying |z - 5| ≤ 3 and having maximum positive principal argument. T

Q: Wait, why do we reject the negative root when calculating the length of OP?

That is a fantastic observation! In the context of the Pythagorean theorem, $OP^2 = OC^2 - CP^2$, we get $OP^2 = 16$. Mathematically, $OP$ could be $\pm 4$. However, $OP$ represents a physical distance (the modulus of the complex number $z$), and distance must always be a non-negative scalar. Thus, we discard the negative root because a modulus cannot be negative.

Q: Why can't we just use the general equation of a line $y = mx$ and intersect it with the circle?

You absolutely can! That is a perfectly valid algebraic approach. You would set $y = mx$ and substitute it into the circle equation $(x-5)^2 + y^2 = 3^2$. For the line to be tangent, the discriminant of the resulting quadratic equation must be zero. However, using the geometric property of the right-angled triangle is often faster and less prone to calculation errors in a high-pressure exam setting.

Q: Why did we multiply by 5 in the step $5z = 16 + 12i$?

That was a strategic move to simplify the arithmetic! By calculating $5z$ first, we cleared the denominators (since $z = \frac{16}{5} + i\frac{12}{5}$), which made the subsequent evaluation of $|5z - 12|^2$ and $|5iz + 16|^2$ much cleaner. It is a common trick in complex number problems to manipulate the expression to avoid working with fractions until the very last step.

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 :

Options:

(A) 26

(B) 12

(C) 20

(D) 16

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler, to the beautiful world of complex numbers. Today, we are not just solving an equation; we are exploring the geometry of the Argand plane.

Imagine you are standing at the origin $(0,0)$ of a vast, two-dimensional landscape. You are given a condition: $|z - 5| \le 3$.

This means that the complex number $z$ lives within a disk centered at $(5, 0)$ with a radius of $3$. Every point inside or on the boundary of this circle is a candidate for $z$.

We are looking for the point that gives us the maximum positive principal argument. The argument $	heta$ is the angle a vector from the origin makes with the positive real axis.

To maximize this angle, we need to stretch our reach as far as possible in the counter-clockwise direction. This happens exactly when our ray from the origin becomes tangent to the circle in the upper half-plane.

### The Right-Angled Revelation

Let $O$ be the origin, $C$ be the center of our circle at $(5, 0)$, and $P$ be the point where our ray touches the circle. Because $CP$ is a radius and $OP$ is a tangent, the radius is perpendicular to the tangent at the point of contact.

Thus, $	riangle OPC$ is a right-angled triangle with $\angle OPC = 90^\circ$. We know the hypotenuse $OC$ is the distance from the origin to $(5, 0)$, which is $5$.

We know the leg $CP$ is the radius of the circle, which is $3$. By the Pythagorean theorem, the base $OP$ is:

$$OP = \sqrt{OC^2 - CP^2} = \sqrt{5^2 - 3^2} = \sqrt{16} = 4$$

This is a beautiful result! It tells us that the modulus of our complex number $z$ at the point of tangency is exactly $4$.

### Constructing the Complex Number

With the geometry settled, we can find the trigonometric ratios for our angle $	heta$. In $	riangle OPC$, we have:

$$\cos 	heta = \frac{	ext{Base}}{	ext{Hypotenuse}} = \frac{4}{5}, \quad \sin 	heta = \frac{	ext{Perpendicular}}{	ext{Hypotenuse}} = \frac{3}{5}$$

Now, we can write $z$ in its polar form:

$$z = |z|(\cos 	heta + i \sin 	heta) = 4\left(\frac{4}{5} + i \frac{3}{5}ight) = \frac{16}{5} + i \frac{12}{5}$$

To make our calculations easier, we multiply by $5$ to get $5z = 16 + 12i$. This substitution is the key that unlocks the final expression.

### The Algebraic Finale

Now, we face the final challenge: evaluating $34 \left| \frac{5z-12}{5iz+16} ight|^2$. Let us tackle the numerator first.

Substituting $5z = 16 + 12i$, we get:

$$|5z - 12|^2 = |16 + 12i - 12|^2 = |4 + 12i|^2 = 4^2 + 12^2 = 16 + 144 = 160$$

Now for the denominator:

$$|5iz + 16|^2 = |i(5z) + 16|^2 = |i(16 + 12i) + 16|^2 = |16i - 12 + 16|^2 = |4 + 16i|^2 = 4^2 + 16^2 = 16 + 256 = 272$$

Finally, we combine these values:

$$34 	imes \frac{160}{272}$$

Simplifying the fraction $\frac{160}{272}$ by dividing both by $16$ gives us $\frac{10}{17}$. Thus:

$$34 	imes \frac{10}{17} = 2 	imes 10 = \mathbf{20}$$

We have arrived at the destination! The final answer is **20**.

Animated Solution for Mathematics - Complex Numbers: Let $z$ be the complex number satisfying $∣ z - 5∣ \leq 3$ and having maximum positive principal argument. Then $34 \frac{5 z - 12}{5 i z + 16}^{2}$ is equal to :

Select Answer:

Visualized Solution

Visualizing the Region $∣ z - 5∣ \leq 3$

Condition for Maximum Argument

Geometry of the Tangent Point

Calculating Distance from Origin

Trigonometry of the Argument

Constructing the Complex Number $z$

Evaluating the Numerator

Evaluating the Denominator

Final Calculation

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Setup

The Right-Angled Revelation

Constructing the Complex Number

The Algebraic Finale

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