# Let $z_1,z_2,z_3$ be any 3 complex numbers, such that $|z_1|=|z_2|=2, |z_3|=1$, $|\arg(\frac{z_1-z_3}{z_2-z_3})|=\frac{\pi}{2}$

Let $$z_1,z_2,z_3$$ be any three complex numbers, such that $$|z_1|=|z_2|=2, |z_3|=1$$ and $$\left|\arg\left(\frac{z_1-z_3}{z_2-z_3}\right)\right|=\frac{\pi}{2}$$. Then, the difference between the maximum and minimum values of $$|z_1+z_2|$$ is equal to

My Approach:

All I can think about is angle between vector $$z_1-z_3$$ and $$z_2-z_3$$ is $$\frac{\pm\pi}{2}$$

Then I tried to plot them on argand plane but can't think further.

• Very rough sketch of solution. As others have pointed out, we can assume $z_3 = 1$ without loss of generality. Then $z_1, z_2$ are on the circle in the complex plane with radius $2$, such that the angle $z_1$-$z_3$-$z_2$ is a right angle. The value $\left| z_1+z_2 \right|$ is maximized when $z_1, z_2$ are at their closest, and minimized when they are at their furthest. The lines passing through $1$ with arguments $\pi/4$ and $-\pi/4$ intersect the radius-$2$ circle in the appropriate points. Geometry or algebra show that the desired difference is $2$. Commented May 6 at 3:48
• @mathofile Your solution was posted as a picture, but one of the editors deleted it. Try editing your work - include what was in the picture. Then your question will not be in danger of being deleted again. Commented May 7 at 21:32

For simplicity, I use same notation of points and their complex coordinates.

$$z_1, z_2$$ lie on a circle with radius $$2,$$ centered at $$0.$$
$$z_3$$ lies on the unit circle $${C}$$ centered at $$0,$$ and on circle $$C'$$ with diameter $$z_1z_2.$$

WLOG, set $$z_1=2.$$
$$|z_1+z_2|$$ is a diagonal length of the rhombus with vertices $$0, z_1, z_1+z_2, z_2.$$ It is grey one in the picture.
$$|z_1+z_2|$$ is maximum if $$z_1, z_2$$ are as close as possible to each other, e.g. circles $$C, C'$$ touch externally in a single point $$z_3.$$

For minimum, I chose the labeling $$w_1, w_2$$ instead of $$z_1, z_2.$$
To better see the relation between lengths, set $$w_1=z_2.$$
$$|w_1+w_2|$$ is minimum if $$w_1, w_2$$ are as far apart as possible, and $$C$$ touches $$C'$$ internally.

The difference between the maximum and minimum is twice the radius of unit circle. $$|z_1+z_2|-|w_1+w_2|=2$$

• [+1] Good solution with geometrical insight. After correction, I still don't see how my own one (that I have cancelled) cannot give the result. Any idea ? Commented May 6 at 10:00
• @JeanMarie I can't see your solution anymore. If you can, sent it to me please. But I only will be able to check it on Wednesday or later. Commented May 6 at 11:01
• I have undeleted it. Thanks for having a look to it when you have time. Commented May 6 at 11:42

The minimum value for $$|z_1+z_2|$$ is about $$2\sqrt{2}\sqrt{1+\cos\left(4\left(0.997414\right)\right)}$$ which is about $$1.64575362143$$

The maximum value for $$|z_1+z_2|$$ is about $$2\sqrt{2}\sqrt{1+\cos\left(4\left(0.212016\right)\right)}$$ which is about $$3.64575362143$$

I'm getting these numbers from Wolfram1 and Wolfram2

WLOG, let $$(z_1,z_2,z_3)=(2e^{ix},2e^{iy},1)$$

Now, we have $$2e^{ix}-1=±ri(2e^{iy}-1)$$ for some real $$r$$

However, in order for $$(z_1,z_2)$$ to satisfy the condition that $$|z_1+z_2|$$ is extremized with respect to the constraint $$\Re(\frac{z_1-1}{z_2-2})=0$$, it turns out that $$r$$ must be $$1$$

(otherwise $$|z_1+z_2|$$ will fail to be extremized with respect to the constraint $$\Re(\frac{z_1-1}{z_2-1})=0$$)

$$r=1$$ gives us

$$\cos(x)±\sin(y)=\frac{1}{2}$$

and

$$\sin(x)∓\cos(y)=∓\frac{1}{2}$$

Hence my Wolfram calculations

As for the minimum and maximum values themselves, it takes only a little bit of trigonometry to demonstrate that $$|z_1+z_2|=|2e^{ix}+2e^{iy}|=2\sqrt{2+2\cos(x-y)}$$

Extra information

Since the system of equations implies that $$x=2n\pi-y$$, that means that $${z_1}/{z_3}$$ and $${z_2}/{z_3}$$ must be complex conjugates, whether we be in the minimum case or the maximum case.

Additionally, you can actually obtain exact figures for $$z_1$$ and $$z_2$$ in terms of $$z_3$$ in a) the minimum case and b) the maximum case.

In the minimum case for example, we have $$(z_1,z_2,z_3)=(2e^{±i\alpha}z_3,2e^{∓i\alpha}z_3,z_3)$$

where $$\alpha=\arccos(\frac{1-\sqrt{7}}{4})$$

In the maximum case, we have $$(z_1,z_2,z_3)=(2e^{±i\beta}z_3,2e^{∓i\beta}z_3,z_3)$$

where $$\beta=\arccos(\frac{\sqrt{7}+1}{4})$$

(and $$z_3$$ is of course some arbitrary number on the unit circle)

• The exact value of the minimum is $\sqrt{7}-1$ and the exact value of the maximum is $\sqrt{7}+1$. Commented May 6 at 3:49