This problem challenges us to find the area of a triangle formed by the focus of a parabola and the points of tangency of tangents drawn from an external point. To solve this, we will systematically apply properties of parabolas, the concept of a chord of contact, and efficient methods for calculating triangle areas.

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1. Understanding the Parabola and Locating its Focus

The first crucial step is to identify the key parameters of the given parabola, specifically its focal length 'a' and the coordinates of its focus 'F'. This will be one of the vertices of our triangle.

• Given Parabola Equation: We are given the equation of the parabola as $y^2 = 8x$.

• Standard Form Comparison: The standard form of a parabola with its vertex at the origin and its axis along the x-axis is $y^2 = 4ax$. By comparing our given equation to this standard form, we can determine the value of 'a'.

• Determining 'a': Equating the coefficients of $x$ from both equations: $4a = 8$ $a = \frac{8}{4}$ $a = 2$ The parameter $a$ represents the focal length and is fundamental to defining the parabola's features.

• Locating the Focus F: For a parabola of the form $y^2 = 4ax$, the focus $F$ is located at $(a, 0)$. Substituting the value of $a=2$: $F = (2, 0)$ This is our first vertex.

• Tip for Success: Always be careful when extracting 'a' from the parabola equation. A common mistake is to directly assume $a=8$ from $y^2=8x$, instead of $4a=8$.

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2. Finding the Equation of the Chord of Contact PQ

When tangents are drawn from an external point to a parabola, the line segment joining the points of tangency is called the chord of contact. We need to find the equation of this line, as it will help us determine the coordinates of points $P$ and $Q$.

• Key Concept: For a parabola $y^2 = 4ax$, the equation of the chord of contact from an external point $(x_1, y_1)$ is given by the formula $T=0$, which expands to: $yy_1 = 2a(x + x_1)$ This formula is a powerful tool for conics, allowing us to find the line connecting the two points where tangents from an external point touch the curve.

• External Point: The tangents are drawn from the point $P_0(-8, 0)$. So, we have $(x_1, y_1) = (-8, 0)$.

• Substituting Values: We use $a=2$, $x_1 = -8$, and $y_1 = 0$ into the chord of contact formula: $y(0) = 2(2)(x + (-8))$ $0 = 4(x - 8)$ $x - 8 = 0$ $x = 8$ Thus, the equation of the chord of contact $PQ$ is $x = 8$.

• Geometric Insight and Self-Check: Notice that the external point $P_0(-8, 0)$ lies on the x-axis, which is the axis of the parabola $y^2 = 8x$. A significant property of parabolas is that if tangents are drawn from a point on the axis of the parabola, the chord of contact is always perpendicular to the axis. Since the axis is the x-axis, our result $x=8$ (a vertical line) being perpendicular to the x-axis is consistent with this property, confirming our calculation.

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3. Determining the Coordinates of Points P and Q

Now that we have the equation of the chord of contact ($x=8$), we can find the exact coordinates of $P$ and $Q$. These points are simply the intersections of the chord of contact with the parabola itself.

• Method: To find the points of intersection, we substitute the equation of the chord ($x=8$) into the equation of the parabola ($y^2 = 8x$).
• Solving for y: $y^2 = 8(8)$ $y^2 = 64$ Taking the square root of both sides, we get two possible values for $y$: $y = \pm\sqrt{64}$ $y = \pm 8$
• Coordinates of P and Q: Therefore, the coordinates of the points of tangency are: $P = (8, 8)$ $Q = (8, -8)$ (The assignment of $P$ to $(8,8)$ and $Q$ to $(8,-8)$ is arbitrary; they are interchangeable for area calculation.)

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4. Calculating the Area of Triangle PFQ

We now have all three vertices of the triangle $PFQ$:

• Focus $F(2, 0)$
• Point $P(8, 8)$
• Point $Q(8, -8)$

While the general determinant formula for the area of a triangle is always an option, we should look for more efficient methods, especially when the vertices have special arrangements. In this case, points $P$ and $Q$ have the same x-coordinate, which means the side $PQ$ is a vertical line segment. This allows for a very straightforward base-height calculation.

• Choosing the Base PQ: Since $P(8, 8)$ and $Q(8, -8)$ share the same x-coordinate ($x=8$), the line segment $PQ$ is a vertical segment. Its length can be easily calculated as the absolute difference of their y-coordinates: $\text{Base } PQ = |y_P - y_Q| = |8 - (-8)| = |8 + 8| = 16 \text{ units}$

• Determining the Height: The height of the triangle is the perpendicular distance from the third vertex, the focus $F(2, 0)$, to the line containing the base $PQ$. The line containing $PQ$ is the vertical line $x=8$. The perpendicular distance from a point $(x_F, y_F)$ to a vertical line $x=k$ is simply $|x_F - k|$. Here, $x_F = 2$ (from $F(2,0)$) and $k=8$ (from line $x=8$). $\text{Height } h = |x_F - 8| = |2 - 8| = |-6| = 6 \text{ units}$

• Area Calculation: Using the standard formula for the area of a triangle, $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$: $\text{Area}(\triangle PFQ) = \frac{1}{2} \times PQ \times h$ $\text{Area}(\triangle PFQ) = \frac{1}{2} \times 16 \times 6$ $\text{Area}(\triangle PFQ) = 8 \times 6$ $\text{Area}(\triangle PFQ) = 48 \text{ sq. units}$

• Tip for Efficiency: Always check if any side of the triangle is horizontal or vertical. If so, the base-height formula is typically much faster and less prone to calculation errors than the general determinant formula.

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5. Important Concepts and Formulas Used (Recap)

For quick review and reinforcement, here are the core mathematical tools applied in this solution:

• Standard Parabola Equation: $y^2 = 4ax$
  • Its focus is located at $F(a, 0)$.
• Equation of Chord of Contact: For a parabola $y^2 = 4ax$, the chord of contact from an external point $(x_1, y_1)$ is given by $yy_1 = 2a(x + x_1)$.
• Area of a Triangle:
  • When a base is parallel to an axis: $\text{Area} = \frac{1}{2} \times \text{base length} \times \text{perpendicular height}$.
  • General formula for vertices $(x_A, y_A)$, $(x_B, y_B)$, $(x_C, y_C)$: $\text{Area} = \frac{1}{2} |x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)|$

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6. Strategic Advice and Common Pitfalls to Avoid

• Master the Basics: A strong understanding of the standard forms and properties of conics (parabolas, ellipses, hyperbolas) is non-negotiable for JEE. Know your definitions for focus, directrix, axis, vertex, and key formulas.
• Chord of Contact $T=0$: This formula is incredibly versatile for all conics. Understand its application for tangents and chords of contact. Practice deriving it to solidify your understanding.
• Geometric Intuition: Develop the habit of visualizing the problem. Sketching the parabola, the external point, and the chord of contact can often reveal shortcuts or help verify your algebraic results (like the perpendicularity of the chord of contact in this problem).
• Calculation Accuracy: Double-check your arithmetic, especially when dealing with signs and square roots. A small error in 'a' or in coordinates can propagate and lead to an incorrect final answer.
• Time Management: In competitive exams, choosing the most efficient method (like base-height for area calculation here) is crucial. Don't blindly jump to the most general formula if a simpler one applies.

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7. Conclusion and Key Takeaway

This problem is a classic example of applying fundamental concepts of parabolas in coordinate geometry. We systematically:

1. Identified the parabola's focal length ($a=2$) and focus ($F(2,0)$).
2. Used the chord of contact formula ($yy_1 = 2a(x+x_1)$) to determine the equation of the line $PQ$ ($x=8$).
3. Found the coordinates of the points of tangency $P(8,8)$ and $Q(8,-8)$ by intersecting the chord of contact with the parabola.
4. Calculated the area of $\triangle PFQ$ efficiently using the base-height method, which was simplified by the vertical nature of the base $PQ$.

The key takeaway is to build a strong foundation in conic sections, be proficient with standard formulas, and strategically select the most efficient mathematical tools for geometric calculations.

The final answer is $\boxed{\text{48}}$.