To solve the problem in the image, let’s break it down step by step:

Given:

Two complex numbers $z_1$ and $z_2$ are represented graphically in the complex plane.
We are tasked to determine the value of $\frac{z_1}{z_2}$.

Observing the Diagram:

1. $z_1$ lies at an angle $\theta$ above the positive real axis.
2. $z_2$ lies at an angle $\theta$ below the negative real axis.

Thus:

• $z_1$ has a magnitude of $r_1 = 8$ and a positive angle $\theta$ (counterclockwise).
• $z_2$ has a magnitude $r_2 = 4$ and a negative angle $\theta$ (clockwise).

Expressing $z_1$ and $z_2$ in Polar Form:

1. For $z_1$:
   $z_1 = r_1 e^{i\theta} = 8 e^{i\theta}$.

2. For $z_2$:
   $z_2 = r_2 e^{-i\theta} = 4 e^{-i\theta}$.

Computing $\frac{z_1}{z_2}$:

$\frac{z_1}{z_2} = \frac{8 e^{i\theta}}{4 e^{-i\theta}}.$ Simplify the magnitudes and the exponents: $\frac{z_1}{z_2} = \frac{8}{4} \cdot e^{i(\theta - (-\theta))}.$ $\frac{z_1}{z_2} = 2 e^{i(2\theta)}.$

Analyzing the Result:

The term $e^{i(2\theta)}$ represents a rotation in the complex plane by $2\theta$. From the diagram, since $\theta$ corresponds to the angles of $z_1$ and $z_2$, we can conclude that $2\theta = \frac{\pi}{2}$ (90 degrees).

Thus: $e^{i\frac{\pi}{2}} = i.$

Substitute this back: $\frac{z_1}{z_2} = 2 \cdot i = 2i.$

Final Answer:

The value of $\frac{z_1}{z_2}$ is:

(d) $2i$.

Would you like a more detailed explanation of the steps or the polar form? Let me know!

Related Questions:

1. What are the properties of polar form for complex numbers?
2. How do we multiply and divide complex numbers in polar form?
3. Why does $e^{i\frac{\pi}{2}} = i$ in the complex plane?
4. How can we calculate the argument of a complex number?
5. What is the geometric interpretation of the magnitude and argument?

Tip: Remember that in polar form, rotations of $e^{i\theta}$ simplify complex number calculations!