<p>Let P be the point of intersection of the common tangents to the parabola \(y^2 = 12x\) and the hyperbola \(8x^2 - y^2 = 8\). If S and S' denote the foci of the hyperbola where S lies on the positive \(x\)-axis then P divides SS' in a ratio:</p>
Step-by-Step Solution
Key Concept: Find the common tangents to both parabola and hyperbola by using the condition that a line is tangent to both curves simultaneously. The point of intersection P of these tangents can then be used to find the ratio in which P divides the focal chord SS'.
<p><strong>Step 1:</strong> Find tangency condition for parabola y² = 12x. A line y = mx + c is tangent if c = 3/m.</p><p><strong>Step 2:</strong> For hyperbola 8x² - y² = 8, rewrite as x²/1 - y²/8 = 1. Here a² = 1, b² = 8, so tangent condition is: c² = a²m² - b² = m² - 8.</p><p><strong>Step 3:</strong> For common tangent: 3/m = ±√(m² - 8). Squaring: 9/m² = m² - 8, so 9 = m⁴ - 8m², giving m⁴ - 8m² - 9 = 0.</p><p><strong>Step 4:</strong> Let u = m². Then u² - 8u - 9 = 0, so (u - 9)(u + 1) = 0. Thus m² = 9, giving m = ±3.</p><p><strong>Step 5:</strong> For m = 3: c = 3/3 = 1. For m = -3: c = 3/(-3) = -1. Common tangents are y = 3x + 1 and y = -3x - 1.</p><p><strong>Step 6:</strong> Intersection point P: Setting 3x + 1 = -3x - 1, we get 6x = -2, so x = -1/3 and y = 0. Thus P = (-1/3, 0).</p><p><strong>Step 7:</strong> For hyperbola x²/1 - y²/8 = 1: c² = a² + b² = 1 + 8 = 9, so c = 3. Foci are S(3, 0) and S'(-3, 0).</p><p><strong>Step 8:</strong> P divides SS' where S(3, 0) and S'(-3, 0). The ratio PS:PS' = |3 - (-1/3)| : |(-1/3) - (-3)| = 10/3 : 8/3 = 10 : 8 = 5 : 4.</p><p>∴ Answer: A</p>
Correct Answer: A