Parabola y2=16x, a=4, focus at (4,0). Point (16,16) has parameter t1=2.
Other end of focal chord: t2=−1/t1=−1/2, giving (4⋅1/4,8⋅(−1/2))=(1,−4).
P divides the chord in ratio 5:2 internally (two cases):
From (16,16) to (1,−4): P=(737,712), α+β=7.
From (1,−4) to (16,16): P=(782,772), α+β=22.
Minimum value of α+β=7.