<p>Tangents drawn from the point \((-8, 0)\) to the parabola \(y^2 = 8x\) touch the parabola at \(P\) and \(Q\). If \(F\) is the focus of the parabola, then the area of the triangle \(PFQ\) (in sq. units) is equal to</p>
Step-by-Step Solution
Key Concept: Use the chord of contact formula to find points P and Q, then leverage the focus-directrix property and the reflective property of parabolas to calculate the triangle area efficiently.
<p><strong>Step 1:</strong> Identify the parabola y² = 8x. Comparing with y² = 4ax, we get 4a = 8, so a = 2. Thus focus F = (2, 0) and directrix x = -2.</p><p><strong>Step 2:</strong> Find the chord of contact from external point (-8, 0). For parabola y² = 4ax and external point (x₁, y₁), the chord of contact is yy₁ = 2a(x + x₁). Here: y(0) = 2(2)(x - 8), which gives 0 = 4(x - 8). This is incorrect; we use the proper formula: The equation is yy₁ = 2a(x + x₁) where (x₁, y₁) = (-8, 0).</p><p><strong>Step 3:</strong> Correct approach: Tangent at point (t², 2t) on parabola y² = 8x is ty = x + 2t². If this passes through (-8, 0): t(0) = -8 + 2t², so 2t² = 8, giving t² = 4, thus t = ±2.</p><p><strong>Step 4:</strong> Find P and Q: For t = 2: point P = (4, 4). For t = -2: point Q = (4, -4).</p><p><strong>Step 5:</strong> Calculate area of triangle PFQ with vertices P(4, 4), F(2, 0), Q(4, -4):<br/>Area = ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|<br/>= ½|4(0 - (-4)) + 2((-4) - 4) + 4(4 - 0)|<br/>= ½|16 - 16 + 16| = ½(16) = 8</p><p>∴ Answer: C (8 sq. units)</p>
Correct Answer: C