Let E denote the parabola $$y^{2} = 8x$$. Let P = (−2, 4), and let Q and $$Q^{'}$$ be two distinct points on E such that the lines PQ and $$PQ^{'}$$ are tangents to E. Let F be the focus of E. Then which of the following statements is (are) TRUE ?

The parabola is $$y^{2}=8x$$ which can be written as $$y^{2}=4ax$$ with $$a=2$$.
For this parabola: focus $$F=(a,0)=(2,0)$$ and the tangent at the parametric point $$(at^{2},2at)$$ is $$ty=x+at^{2}$$.

Let $$Q(at_{1}^{2},2at_{1})$$ and $$Q'(at_{2}^{2},2at_{2})$$ be the two points of contact of the tangents drawn from the external point $$P=(-2,4)$$.
Put $$x=-2,\;y=4$$ in the tangent equation $$ty=x+2t^{2}$$:

$$4t=-2+2t^{2}\;\Longrightarrow\;2t^{2}-4t-2=0$$
$$\Longrightarrow\;t^{2}-2t-1=0\;\Longrightarrow\;t=1\pm\sqrt{2}$$

Hence,
$$t_{1}=1+\sqrt{2},\quad t_{2}=1-\sqrt{2}$$

Coordinates of the points of contact:

$$Q=(2t_{1}^{2},\,4t_{1})=\bigl(6+4\sqrt{2},\,4+4\sqrt{2}\bigr)$$
$$Q'=(2t_{2}^{2},\,4t_{2})=\bigl(6-4\sqrt{2},\,4-4\sqrt{2}\bigr)$$

Checking statement C
Distance $$PF=\sqrt{(2-(-2))^{2}+(0-4)^{2}}=\sqrt{4^{2}+(-4)^{2}}=\sqrt{32}=4\sqrt{2}$$.
Given claim is $$5\sqrt{2}$$, so statement C is FALSE.

Checking statement D
Slope of $$QQ'$$: $$m_{QQ'}=\frac{(4+4\sqrt{2})-(4-4\sqrt{2})}{(6+4\sqrt{2})-(6-4\sqrt{2})}=1$$
Equation of $$QQ'$$ (through $$Q$$) is $$y=(x-2)$$.
Focus $$F(2,0)$$ satisfies $$0=2-2$$, so $$F$$ lies on $$QQ'$$. Statement D is TRUE.

Checking statement B
Slopes of the two tangents:

$$m_{PQ}=\frac{4+4\sqrt{2}-4}{6+4\sqrt{2}+2}=\frac{4\sqrt{2}}{8+4\sqrt{2}}=\frac{\sqrt{2}}{2+\sqrt{2}}$$
$$m_{PQ'}=\frac{4-4\sqrt{2}-4}{6-4\sqrt{2}+2}=\frac{-4\sqrt{2}}{8-4\sqrt{2}}=\frac{-\sqrt{2}}{2-\sqrt{2}}$$

$$m_{PQ}\,m_{PQ'}=\frac{\sqrt{2}}{2+\sqrt{2}}\cdot\frac{-\sqrt{2}}{2-\sqrt{2}}=-1$$
Since the product of slopes is $$-1$$, $$PQ\perp PQ'$$, so $$\triangle QPQ'$$ is right-angled at $$P$$. Statement B is TRUE.

Checking statement A
Slope $$PF=-1$$ and slope $$FQ=1$$, hence $$PF\perp FQ$$. Therefore $$\triangle PFQ$$ is right-angled at $$F$$. Statement A is TRUE.

Thus the correct statements are:
Option A (triangle PFQ is right-angled),
Option B (triangle QPQ' is right-angled),
Option D (F lies on line QQ').

Answer: Option A, Option B, Option D.