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? (1) The triangle $PFQ$ is a right-angled triangle (2) The triangle $QPQ'$ is a right-angled triangle (3) The distance between $P$ and $F$ is $5\sqrt{2}$ (4) $F$ lies on the line joining $Q$ and $Q'$

Let E denote the parabola y² = 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? $\text{Given: } E:\ y^2 = 8x$ $P(-2,4);\quad PQ,\ PQ' \text{ are tangents to } E$ $F = \text{the focus of } E$ $\text{To find: which of } (1),\ (2),\ (3),\ (4) \text{ are TRUE}$ $\text{Plan: find } F, \text{ tangents from } P, \text{ then test each statement}$ $(1)\ PFQ\quad (2)\ QPQ'\quad (3)\ PF\quad (4)\ F \in QQ'$ $y^2 = 4ax \Rightarrow 4a = 8 \Rightarrow a = 2$ $F = (a,0) = (2,0)$ $\text{Parabola } y^2 = 8x,\ \text{focus } F(2,0)$ $\text{Tangent of slope } m:\ y = mx + \dfrac{a}{m} = mx + \dfrac{2}{m}$ $\text{At } P(-2,4):\ 4 = -2m + \dfrac{2}{m}$ $m^2 + 2m - 1 = 0 \Rightarrow m = -1 \pm \sqrt{2}$ $\text{Contact}: \left(\dfrac{a}{m^2}, \dfrac{2a}{m}\right) = \left(\dfrac{2}{m^2}, \dfrac{4}{m}\right)$ $Q\left(6+4\sqrt{2},\ 4+4\sqrt{2}\right)$ $Q'\left(6-4\sqrt{2},\ 4-4\sqrt{2}\right)$ $\text{Tangents } PQ \text{ and } PQ'$ $\overrightarrow{FP} = (-4,4),\quad \overrightarrow{FQ} = (4+4\sqrt{2},\ 4+4\sqrt{2})$ $\overrightarrow{FP}\cdot\overrightarrow{FQ} = -4(4+4\sqrt{2}) + 4(4+4\sqrt{2}) = 0$ $\Rightarrow \angle PFQ = 90^\circ \Rightarrow (1)\ \text{TRUE}$ $\overrightarrow{PQ} = 4(2+\sqrt{2},\ \sqrt{2}),\quad \overrightarrow{PQ'} = 4(2-\sqrt{2},\ -\sqrt{2})$ $\overrightarrow{PQ}\cdot\overrightarrow{PQ'} = 16\left[(4-2) - 2\right] = 0$ $\Rightarrow \angle QPQ' = 90^\circ \Rightarrow (2)\ \text{TRUE}$ $(3)\ PF = \sqrt{(2-(-2))^2 + (0-4)^2} = \sqrt{16+16}$ $= 4\sqrt{2} \ne 5\sqrt{2} \Rightarrow (3)\ \text{FALSE}$ $(4)\ \text{Chord of contact } QQ':\ y = x - 2$ $F(2,0):\ 0 = 2 - 2\ \checkmark \Rightarrow (4)\ \text{TRUE}$ $(1),\ (2),\ (4)\ \text{are TRUE};\quad (3)\ \text{is FALSE}$ $\boxed{\text{Answer: } (1),\ (2),\ (4)}$