How to prove that the feasible set of a two-asset portfolio is a hyperbola?

The question comes from ‘Mathematics for Finance: An Introduction to Financial Engineering’ by Marek Capiński (Author), Tomasz Zastawniak. The book does not give a complete proof, and I did not find a detailed proof online. But I think this issue is still very important. So posted the question on the stackexchange.

Assume that $$-1<\rho_{12}<1$$ ， and $$\mu_1\neq\mu_2$$ Then for each asset portfolio V in the feasible set,Prove $$x=\sigma_{v}$$ and $$y=\mu_{v}$$ satisfy the hyperbolic equation $$x^2-A^2(y-\mu_0)^2=\sigma_0^2$$ $$A^2=\frac{\sigma_1^2+\sigma_2^2-2c_{12}}{\left(\mu_1-\mu_2\right)^2}>0$$The two asymptotes of this hyperbola are $$y=\mu_0\pm\frac1Ax$$

Background Information：

Then $$\sigma_v^2$$ as a function of s reaches the minimum value at the following point $$s_0$$ $$s_0=\frac{\sigma_2^2-c_{12}}{\sigma_1^2+\sigma_2^2-2c_{12}}$$ Because$$\frac{\mathrm{d}(\sigma_V^2)}{\mathrm{d}s}=2s(\sigma_1^2+\sigma_2^2-2c_{12})-2(\sigma_2^2-c_{12})=0$$ Where$$\mu_0=\frac{\mu_1\sigma_2^2+\mu_2\sigma_1^2-(\mu_1+\mu_2)c_{12}}{\sigma_1^2+\sigma_2^2-2c_{12}}$$$$\sigma_0^2=\frac{\sigma_1^2\sigma_2^2-c_{12}}{\sigma_1^2+\sigma_2^2-2c_{12}}$$ $$\sigma_1$$ is the standard deviation of asset 1’s rate of return

$$\sigma_2$$ is the standard deviation of asset 2’s rate of return

$$\mu_1$$ is the expected return on asset 1

$$\mu_2$$ is the expected return on asset 2

$$c_{12}$$ is the covariance of the returns on Asset 1 and Asset 2

expected value of portfolio v:$$\\;\mu_v=s\mu_1+(1-s)\mu_2$$

Variance of portfolio v:$$\\;\sigma_v^2=s^2\sigma_1^2+(1-s)^2\sigma_2^2+2s(1-s)c_{12}\\$$ Formula 3.9

The book gives simple proof tips：Substituting $$s=\frac{\mu_V-\mu_2}{\mu_1-\mu_2}$$ into Equation 3.9.Using simple (or can be considered slightly cumbersome) transformations, we can prove that the proposition is true

geogebra diagram