If $\mathrm{a}={\mathrm{xcos}}^2\mathrm{\alpha }+\mathrm{y} {\sin }^2 \mathrm{\alpha }$ then $\left(\mathrm{x}-a\right)\left(\mathrm{y}-\mathrm{a}\right)+{\left(\mathrm{x}-\mathrm{y}\right)}^2 {\sin }^2\mathrm{\alpha } {\cos }^2\mathrm{\alpha }=$

$\begin{array}{lcl}\mathrm{a}\begin{array}{cl}=& \mathrm{x} {\cos }^2\mathrm{\alpha }+{\mathrm{ysin}}^2\mathrm{\alpha }\end{array}& & \\ \mathrm{a}\begin{array}{cl}=& \mathrm{x}-\mathrm{x} {\sin }^2\mathrm{\alpha }+\mathrm{y} {\sin }^2\mathrm{\alpha }\end{array}& & \\ (\mathrm{x}-\mathrm{y}){\sin }^2\mathrm{\alpha }\begin{array}{cl}=& \mathrm{x}-\mathrm{a}\end{array}& & \\ \mathrm{a}\begin{array}{cl}=& \mathrm{x} {\cos }^2\mathrm{\alpha }+\mathrm{y}-\mathrm{y} {\cos }^2\mathrm{\alpha }\end{array}& & \\ \mathrm{y}-\mathrm{a}\begin{array}{cl}=& -(\mathrm{x}-\mathrm{y}){\cos }^2\mathrm{\alpha }\end{array}& & \\ (\mathrm{x}-\mathrm{a})(\mathrm{y}-\mathrm{a})\begin{array}{cl}=& -{(\mathrm{x}-\mathrm{y})}^2 {\sin }^2\mathrm{\alpha } {\cos }^2\mathrm{\alpha }\end{array}& & \\ (\mathrm{x}-\mathrm{a})(\mathrm{y}-\mathrm{a})+{(\mathrm{x}-\mathrm{y})}^2 {\sin }^2\mathrm{\alpha } {\cos }^2\mathrm{\alpha }& =& 0\\ & & \end{array}$