Step 1: Analyze the Parabola and Given Points

The equation of the parabola E is given as $y^2 = 8x$. This is a standard parabola of the form $y^2 = 4ax$. Comparing the two equations, we get 4a = 8, which implies a = 2.

• The focus F of the parabola is at (a, 0), so F = (2, 0).
• The vertex is at the origin (0, 0).
• The equation of the directrix is x = -a, which is x = -2.

The given point is P = (-2, 4).

Step 2: Determine the Location of Point P

By observing the coordinates of point P (-2, 4), we can see that its x-coordinate is -2. This means that point P lies on the directrix of the parabola, x = -2.

This observation is key, as several properties of a parabola are related to its directrix.

Step 3: Evaluate each statement

A: The triangle PFQ is a right-angled triangle

There is a standard property of parabolas: The portion of a tangent between the point of contact and the directrix subtends a right angle at the focus.

In this case, PQ is a tangent to the parabola, where Q is the point of contact and P is a point on the directrix. According to this property, the line segment PQ must subtend a right angle at the focus F. This means that the angle ∠PFQ = 90°. Therefore, the triangle PFQ is a right-angled triangle (right-angled at F).

Alternatively, using slopes: Let the coordinates of Q be $(at^2, 2at) = (2t^2, 4t)$. The equation of the tangent at Q is $yt = x + at^2$, which is $yt = x + 2t^2$. Since this tangent passes through P(-2, 4), we have $4t = -2 + 2t^2$, which simplifies to $t^2 - 2t - 1 = 0$. Let $t_1$ be the parameter for point Q. Slope of PF: $m_{PF} = (4-0)/(-2-2) = 4/(-4) = -1$. Slope of FQ: $m_{FQ} = (4t_1 - 0) / (2t_1^2 - 2) = 4t_1 / (2(t_1^2 - 1)) = 2t_1 / (t_1^2 - 1)$. For ∠PFQ = 90°, we must have $m_{PF} * m_{FQ} = -1$. $(-1) * (2t_1 / (t_1^2 - 1)) = -1$ $2t_1 = t_1^2 - 1$ $t_1^2 - 2t_1 - 1 = 0$. This is the same condition that $t_1$ must satisfy for the tangent to pass through P. Thus, the condition is met. Statement A is TRUE.

B: The triangle QPQ' is a right-angled triangle

Another standard property of parabolas is that tangents drawn from any point on the directrix are perpendicular to each other.

Since P (-2, 4) lies on the directrix, the two tangents PQ and PQ' must be perpendicular. This means the angle ∠QPQ' = 90°. Therefore, the triangle QPQ' is a right-angled triangle (right-angled at P).

Alternatively, using parameters: From the quadratic $t^2 - 2t - 1 = 0$, let the roots be $t_1$ and $t_2$, corresponding to points Q and Q'. The slope of the tangent at $(2t^2, 4t)$ is 1/t. So the slopes of tangents PQ and PQ' are $m_1 = 1/t_1$ and $m_2 = 1/t_2$. The product of the slopes is $m_1 m_2 = (1/t_1)(1/t_2) = 1/(t_1 t_2)$. From Vieta's formulas for the quadratic, the product of roots $t_1 t_2 = -1$. So, $m_1 m_2 = -1$. The tangents are perpendicular. Statement B is TRUE.

C: The distance between P and F is 5√2

The coordinates are P (-2, 4) and F (2, 0). We use the distance formula: $PF = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ $PF = \sqrt{(2 - (-2))^2 + (0 - 4)^2}$ $PF = \sqrt{(4)^2 + (-4)^2}$ $PF = \sqrt{16 + 16} = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$.

The statement says the distance is $5\sqrt{2}$, which is incorrect. Statement C is FALSE.

D: F lies on the line joining Q and Q'

A third standard property of parabolas is that the chord of contact of tangents drawn from any point on the directrix is a focal chord (i.e., it passes through the focus).

The line joining Q and Q' is the chord of contact for tangents drawn from point P. Since P lies on the directrix, the chord of contact QQ' must pass through the focus F. Therefore, F lies on the line joining Q and Q'.

Alternatively, using chord of contact equation: The equation of the chord of contact from an external point $(x_0, y_0)$ to the parabola $y^2 = 4ax$ is $yy_0 = 2a(x + x_0)$. For P (-2, 4) and a=2, the equation of QQ' is: y(4) = 2(2)(x + (-2)) 4y = 4(x - 2) y = x - 2. To check if the focus F (2, 0) lies on this line, substitute its coordinates into the equation: 0 = 2 - 2, which is 0 = 0. The condition is satisfied. Statement D is TRUE.

Conclusion

Based on the analysis, statements A, B, and D are true, while statement C is false.