Let F be the focus of the parabola $y^2 = 8x$. From a point A on the parabola, drop a perpendicular to the directrix of the parabola, with the foot of the perpendicular being B. Let C and D be the two points where the line BF intersects the parabola. When $\overline{\mathrm{BC}} = \overline{\mathrm{CD}}$, what is the area of triangle ABD? (where $\overline{\mathrm{CF}} < \overline{\mathrm{DF}}$ and point A is not the origin) [3 points] (1) $100\sqrt{2}$ (2) $104\sqrt{2}$ (3) $108\sqrt{2}$ (4) $112\sqrt{2}$ (5) $116\sqrt{2}$
Let F be the focus of the parabola $y^2 = 8x$. From a point A on the parabola, drop a perpendicular to the directrix of the parabola, with the foot of the perpendicular being B. Let C and D be the two points where the line BF intersects the parabola. When $\overline{\mathrm{BC}} = \overline{\mathrm{CD}}$, what is the area of triangle ABD? (where $\overline{\mathrm{CF}} < \overline{\mathrm{DF}}$ and point A is not the origin) [3 points]\\
(1) $100\sqrt{2}$\\
(2) $104\sqrt{2}$\\
(3) $108\sqrt{2}$\\
(4) $112\sqrt{2}$\\
(5) $116\sqrt{2}$