You dont need equation for chord of contact
Simply,
dy/dx = x/4
tangent at point (t,t^2/8)
y - t^2/8 = t/4(x-t)
8y - t^2 = 2t(x-t) goes through (m,n)
8n - t^2 = 2tm - 2t^2
t^2 - 2tm + 8n = 0
Since t is the x coordinate,
x^2 - 2mx + 8n = 0 has roots that are the x-coordinates of A,B