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