If any line through (0, 0) cuts the parabola y² = 8(x+2) at A and B. If the circle on AB as diameter always touches $\mathrm{\alpha x}+\mathrm{\beta y}+4=0$ then $2\mathrm{\alpha }+\mathrm{\beta }+4=$

(0, 0) is focus of the parabola. The circle on focal chord as diameter always touches directrix.

$$\begin{gathered} where a=2 and vertex=(h,k)=(-2,0) \\ equation of directrix is x-h+a=0 \\ \Rightarrow x+2+2=0 \\ \Rightarrow x+4=0 \\ compare with \alpha x+\beta +4=0\Rightarrow \alpha =1,\beta =0 \\ Now 2\alpha +\beta +4=6 \end{gathered}$$