<p>If the curves \(y^2 = 6x\), \(9x^2 + by^2 = 16\) intersect each other at right angles, then the value of \(b\) is</p>
Step-by-Step Solution
Key Concept: Two curves intersect at right angles when their tangents at the point of intersection are perpendicular. Find the slopes of tangents to both curves at their intersection point and apply the condition m₁·m₂ = -1.
<p><strong>Step 1:</strong> Find the slopes of tangents to both curves.</p><p>For parabola y² = 6x: Differentiating, 2y(dy/dx) = 6 ⟹ dy/dx = 3/y</p><p>For ellipse 9x² + by² = 16: Differentiating, 18x + 2by(dy/dx) = 0 ⟹ dy/dx = -9x/(by)</p><p><strong>Step 2:</strong> Apply perpendicularity condition at intersection point (x₀, y₀).</p><p>At intersection: m₁ · m₂ = -1</p><p>(3/y₀) · (-9x₀/(by₀)) = -1</p><p>-27x₀/(by₀²) = -1</p><p>27x₀ = by₀²</p><p><strong>Step 3:</strong> Use the fact that (x₀, y₀) lies on the parabola.</p><p>From y₀² = 6x₀, we have: 27x₀ = b(6x₀)</p><p>27x₀ = 6bx₀</p><p>Since x₀ ≠ 0 at intersection: 27 = 6b</p><p>b = 27/6 = 9/2</p><p><strong>Step 4:</strong> Verify by checking the point lies on both curves with b = 9/2.</p><p>∴ Answer: b = 9/2</p>
Correct Answer: C