Let z satisfies  $z^3={\alpha }^3{z}_1^3+{(1-\alpha )}^3{z}_2^3+3(\alpha -{\alpha }^2)z{z}_1{z}_2$ and $f:R\to [0,\infty ),f\left(\alpha \right)=|z{|}^2$, find minimum value of $f\left(\alpha \right)$ given that $z_1=3+4i$  and  $z_2=1-2i$

$z^3+{(-\alpha z_1)}^3+{(-(1-\alpha )z_2)}^3=3z(-\alpha z_1)(-(1-\alpha )z_2)$

$\Rightarrow z-\alpha z_1-(-1-\alpha )z_2=0,\alpha \in R$

So, $\left|z\right|$ is perpendicular distance from  $(0,0)$ to line joining the points $(3,4)$  and $(1,-2)$