Given a measurement operator in the general Hemitian form $$ M = \begin{pmatrix} z_1 & x+iy \\ x-iy & z_2\end{pmatrix}, $$ where $x,y,z_1,z_2 \in \mathbb{R}$, show that the eigenvalues are $$ m_{1,2} = \frac{z_1 + z_2}{2} \pm \frac{1}{2}\Big[ (z_1-z_2)^2 + 4(x^2+y^2)\Big]^{1/2} $$ with eigenvectors $$ |m_1\rangle = \begin{pmatrix}e^{i\varphi}\cos(\theta/2) \\ \sin(\theta/2)\end{pmatrix} $$ and $$ |m_2\rangle = \begin{pmatrix}-e^{i\varphi}\sin(\theta/2) \\ \cos(\theta/2)\end{pmatrix}, $$ where $\varphi = \tan^{-1}(y/x)$ and $\theta = \cos^{-1}([z_1-z_2]/[m_1-m_2])$.
I have found the eigenvalues but I am not able to find eigenstates in that form after hours of work.