Let PQ be the focal chord of the parabola y^2=8x with P=(8,8), where F is the focus.

The parabola $y^2=8x$ has $a=2$, focus $F(2,0)$ and directrix $x=-2$; points are $(2t^2,4t)$. $P$ corresponds to $t=2$. For a focal chord the parameters satisfy $tt'=-1$, so $t'=-\frac12$ and $Q=(\frac12,-2)$. (A) True: the focal chord length is $x_P+x_Q+2a=8+\frac12+4=\frac{25}{2}$. (B) False: the area of $\triangle OPQ$ is $\frac12|8(-2)-\frac12\cdot8|=\frac12\cdot20=10$, not $12$. (C) True: the circle on $PQ$ as diameter has centre $\left(\frac{17}{4},3\right)$ and radius $\frac{25}{4}$; its distance from the line $x=-2$ is $\frac{17}{4}+2=\frac{25}{4}$, so it touches the directrix (a general property for focal chords). (D) True: tangent slopes at parameter $t$ is $\frac1t$, so they are $\frac12$ and $-2$, whose product is $-1$, hence the tangents at $P$ and $Q$ are perpendicular (and meet on the directrix). Correct options: A, C, D.