In a right angled triangle, its two medians are members of the family  $(7x+y-16)+\lambda (2x+5y-14)=0$ and its two altitudes are members of the family $(4x-y-28)+\mu (x-3y+26)=0$, then

Centroid is intersection point of medians$\Rightarrow (2,2)$

Orthocenter is intersection point of attitudes$\Rightarrow (10,12)$

$\frac{2\alpha +10}{3}=2 \frac{2\beta +12}{3}=2$

$\alpha =-2 \beta =-3$

Circumcentre (–2, –3)

So the triangle is variable and area of triangle is maximum when hypotenuse will be perpendicular to the line joining orthocenter and circumcentre.