A and C are two points in an Argand diagram representing the complex numbers -5+2i and 1+I respectively. Given that AC is the diagonal of the square ABCD, find the complex numbers represented by the points B and D.

Answer: -5/2 -3/2 i ;-3/2 +9/2i

Relevance
• Ian H
Lv 7
1 month ago

Lots of ways to do this; continue with Cartesian A(-5, 2), and C(1, 1)

Interpret AC as diameter of length √(6^2 + 1)

Centre is at (-2, 3/2) radius is √(37)/2 so equation is

(x + 2)^2 + (y – 3/2)^2 = 37/4

Slope of AC is -1/6, so, perpendicular diameter will have slope 6

y = 6x + c through centre

3/2 = -12 + c, so equation is y = 6x + 27/2 and they look like this

https://www.wolframalpha.com/input/?i=%28x+%2B+2%2...

y – 3/2 is 6x + 12, so on substitution we have

(x + 2)^2 + 36(x + 2)^2 = 37/4

(x + 2)^2 = 1/4 with solutions

x = -3/2 and x = -5/2 and then

y = 9/2 and y = -3/2 so returning to the complex plane

B and D are 3/2 +9/2i and -5/2 -3/2 i

• Pope
Lv 7
1 month ago

The Argand plane is really only the Cartesian plane used to represent complex numbers. In this plane, x represents the real part of a complex number, and y represents the coefficient of the imaginary part. Here are points A and C expressed as coordinates:

A(-5, 2), C(1, 1)

Now you are working with real coordinates. These are opposite vertices of a square. Find the coordinates of the other two vertices, then write each of those ordered pairs as the two parts of a complex number.

I do not want to be a scold now, but I have left this as an elementary coordinate geometry problem. If you cannot derive those two required coordinate pairs, then let me suggest that you do not have the prerequisites for the study of complex analysis.