If a ≠ 0 and the line 2 b x+3 c y+4 d=0 passes through the points...

Points of intersection of given parabolas are $(0,0)$ and $(4 a, 4 a)$ $\Rightarrow$ equation of line passing through these points is $y=x$ On comparing this line with the given line $2 b x+3 c y+4 d=0$, we get $d=0$ and $2 b+3 c=0 \Rightarrow(2 b+3 c)^2+d^2=0$