review the graph.
what is the difference between ( z_1 ) and ( z_2 ), and where is the difference located?
$ \circ $ difference: ( 0 + 4i ); location: on the real axis
$ \circ $ difference: ( -10 + 0i ); location: on the real axis
$ \circ $ difference: ( 0 + 4i ); location: on the imaginary axis
$ \circ $ difference: ( -10 + 0i ); location: on the imaginary axis

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# Brief Explanations:
1. First, determine the complex numbers $ z_1 $ and $ z_2 $ from the graph. Assume $ z_1 $ is at $ -5 + 2i $ (left on real axis, $ y = 2 $) and $ z_2 $ is at $ 5 + 2i $ (right on real axis, $ y = 2 $).
2. Calculate the difference $ z_1 - z_2 = (-5 + 2i)-(5 + 2i)=-5 - 5+2i - 2i=-10 + 0i $. Wait, no, maybe another approach. Wait, maybe the imaginary parts are same? Wait, looking at the graph, $ z_1 $ and $ z_2 $ have same imaginary component (since they are on the same horizontal line, $ y = 2 $). Wait, maybe I misread. Wait, the options have differences with imaginary part 4 or real part -10. Wait, maybe $ z_1 $ is at $ -5 + 2i $ and $ z_2 $ is at $ 5 + 2i $, then $ z_1 - z_2=-10 + 0i $, which is on the real axis (since imaginary part is 0). Alternatively, if $ z_1 $ and $ z_2 $ have real parts same and imaginary parts differ? Wait, no, the options: let's re - evaluate. Wait, maybe the complex numbers are $ z_1=a + bi $ and $ z_2=c + di $. From the graph, if $ z_1 $ is at, say, $ -5 + 2i $ and $ z_2 $ is at $ 5 + 2i $, then $ z_1 - z_2=(-5 - 5)+(2 - 2)i=-10+0i $, which is on the real axis (because imaginary part is 0). Now check the options: the option with difference $ -10 + 0i $ and location on real axis is one of the options. Wait, but also, if the imaginary parts are different: suppose $ z_1=x + 2i $ and $ z_2=x + 6i $, then $ z_1 - z_2=0-4i $ or $ z_2 - z_1=0 + 4i $. Wait, maybe I made a mistake earlier. Let's look at the y - axis (imaginary axis) labels: 2,4,6. So if $ z_1 $ is at, say, $ x + 2i $ and $ z_2 $ is at $ x + 6i $, then $ z_2 - z_1=(x - x)+(6 - 2)i = 0+4i $, which is on the imaginary axis (since real part is 0). Wait, now I'm confused. Wait, the graph: the imaginary axis is vertical. Let's assume the two points $ z_1 $ and $ z_2 $ have the same real part (so they are vertically aligned) and different imaginary parts. So $ z_1 = a+bi $, $ z_2=a + di $, then $ z_1 - z_2=(a - a)+(b - d)i=(b - d)i $. If $ b - d=-4 $, then $ z_1 - z_2=0-4i $, but the options have $ 0 + 4i $, so maybe $ z_2 - z_1=0 + 4i $. And since the real part is 0, the number is on the imaginary axis (because a complex number $ 0+4i $ is of the form $ 0+yi $, so it lies on the imaginary axis). Wait, now I think I messed up the first approach. Let's start over.

Complex numbers on the complex plane: real part is x - axis, imaginary part is y - axis. So a complex number $ z=x + yi $. If two complex numbers have the same real part (x - coordinate), then their difference will have real part 0 (since $ (x + y_1i)-(x + y_2i)=(x - x)+(y_1 - y_2)i=(y_1 - y_2)i $), so the difference is a pure imaginary number, lying on the imaginary axis. If two complex numbers have the same imaginary part (y - coordinate), their difference will have imaginary part 0, lying on the real axis.

Looking at the graph, the two points $ z_1 $ and $ z_2 $: let's assume $ z_1 $ is at, say, $ x_1 + y_1i $ and $ z_2 $ is at $ x_2 + y_2i $. From the options, the difference with $ 0+4i $ has real part 0, so the two numbers must have the same real part (so $ x_1=x_2 $) and different imaginary parts ($ y_1 - y_2 = 4 $ or $ y_2 - y_1 = 4 $). So the difference is $ 0+4i $ (if $ y_2 - y_1 = 4 $), and since the real part is 0, it's on the imaginary axis.

Now, check the options: the option "difference: $ 0 + 4i $; location: on the imaginary axis" is one of the options. Let's verify: if $ z_1=a + bi $ and $ z_2=a + (b - 4)i $, then $ z_2 - z_1=(a - a)+((b - 4)-b)i=-4i $, no. Wait, $ z_1=a + (b - 4)i $ and $ z_2=a + bi $, then $ z_2 - z_1=0 + 4i $, which is on the imaginary axis (since real part is 0). So this makes sense. So the difference is $ 0 + 4i $ and it's on the imaginary axis.

# Answer:
C. difference: $ 0 + 4i $; location: on the imaginary axis (assuming the option labels are A, B, C, D with C being "difference: $ 0 + 4i $; location: on the imaginary axis")

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