# Given $\frac{z_1}{2z_2}+\frac{2z_2}{z_1} = i$ and $0, z_1, z_2$ form two triangles with $A, B$ the least angles of each. Find $\cot A +\cot B$

Question: If $$z_1$$ and $$z_2$$ are two complex numbers satisfying $$\frac{z_1}{2z_2}+\frac{2z_2}{z_1} = i$$ and $$0, z_1, z_2$$ form two non-similar triangles. $$A, B$$ are the least angles in the two triangles, then $$\cot A +\cot B$$ equals:

I tried solving the first equation by trying to complete the square but to no avail. Then I tried taking $$\frac{z_1}{z_2}$$ as another variable $$z$$, hoping to use the rotation method but I couldn't figure out what to do with it. I think that I'm missing something, but even if I calculate $$z_1$$ and $$z_2$$ then would it not be insufficient to find a condition for minimum values of the cotangents of the angles?

• I can't see how three points: $0, z_1, z_2$ are going to form two triangles. Commented Jul 22, 2020 at 14:31
• "And 0,z1,z2 form two non-similar triangles. " What the heck can that possibly mean?.. Oh... Are they saying $0, z_1, Re(z_1)$ forms one right triangle and $0, z_2, R(z_2)$ forms another? Commented Jul 22, 2020 at 15:56

Write the equation $$\frac{z_1}{2z_2}+\frac{2z_2}{z_1} = i$$ as $$(\frac{z_1}{2z_2})^2 -i \frac{z_1}{2z_2} +1 =0$$ and solve to obtain

$$\left(\frac{z_1}{z_2}\right)_{1,2}=(\sqrt5+1)e^{i\frac\pi2},\> (\sqrt5-1)e^{i\frac{3\pi}2}$$

Thus, the two are right triangles with $$|z_1|>|z_2|$$ and

$$\cot A + \cot B= \left|\left(\frac {z_2}{z_1}\right)_1\right|+ \left|\left(\frac {z_2}{z_1}\right)_2\right|= {\sqrt5+1}+ {\sqrt5-1}=2{\sqrt5}$$

• Thanks, I totally missed that part could be converted into a simple form by squaring both sides. Also, how can one judge that this is the least value?
– sfsg
Commented Jul 22, 2020 at 15:44
• @user807908 - the least angle is determined by $|z_1|>|z_2|$, i.e. the leg $z_1$ is longer Commented Jul 22, 2020 at 16:09

Number one rule: get rid of your denominator by multiplying your equation by z1 * z2. Now, you should have just a numerator, right? Next, separate real and imaginary parts and solve the two parts as equations: f(reals) = 0 and g(imag) = 1. Got it?

Hope this helps.