Official solution
Slope of line y = x is 1
Line AB is perpendicular to line y = x so
Slope of AB = -1
Also slope of AB = \[{{\alpha - \beta } \over {2\alpha - \beta }}\]
\[ \therefore \] \[{{\alpha - \beta } \over {2\alpha - \beta }}\] = -1
\[ \Rightarrow \] 3\[\alpha \] = 2\[\beta \]
h = \[{{2\alpha + \beta } \over 2}\]
\[ \Rightarrow \] 2h = \[{{4\alpha + 2\beta } \over 2}\] = \[{{4\alpha + 3\alpha } \over 2}\] = \[{{7\alpha } \over 2}\]
Also k = \[{{\alpha + \beta } \over 2}\]
\[ \Rightarrow \] 2k = \[{{5\alpha } \over 2}\]
So \[{h \over k} = {7 \over 5}\]
\[ \Rightarrow \] 5h = 7k
\[ \Rightarrow \] 5x = 7y