<p>The equation of parabola is \(y^2 = 12x\) and the equation of hyperbola is \(8x^2 - y^2 = 8\). A common tangent to both curves is drawn. If the tangents \(y = 3x + 1\) and \(y = -3x - 1\) intersect at point P, and S, S' are foci of the hyperbola, then the ratio \(\dfrac{SP}{PS'}\) is:</p>
Step-by-Step Solution
Key Concept: A common tangent to both parabola and hyperbola must satisfy both curve equations simultaneously. The ratio SP/PS' can be found using the focal property of hyperbola: for any point P, ||PS| - |PS'|| = 2a, combined with the geometry of the intersection point of two tangent lines.
<p><strong>Step 1:</strong> Identify point P by finding intersection of tangents y = 3x + 1 and y = -3x - 1.</p><p>Setting equal: 3x + 1 = -3x - 1 → 6x = -2 → x = -1/3</p><p>Then y = 3(-1/3) + 1 = 0, so P = (-1/3, 0)</p><p><strong>Step 2:</strong> Verify these are common tangents. For parabola y² = 12x, tangent at parameter t is ty = x + 3t². For y = 3x + 1: comparing gives 3t = 1 and 3t² = 1, so t = 1/3 ✓</p><p><strong>Step 3:</strong> Find foci of hyperbola 8x² - y² = 8. Rewrite as x²/1 - y²/8 = 1, so a² = 1, b² = 8.</p><p>Then c² = a² + b² = 9, so c = 3. Foci: S = (3, 0) and S' = (-3, 0)</p><p><strong>Step 4:</strong> Calculate distances from P = (-1/3, 0):</p><p>SP = |3 - (-1/3)| = 10/3</p><p>PS' = |(-3) - (-1/3)| = 8/3</p><p><strong>Step 5:</strong> Find ratio: SP/PS' = (10/3)/(8/3) = 10/8 = 5/4</p><p>∴ Answer: A</p>
Correct Answer: A