<p>We have the curves: \(x^2 + y^2 = 9\) and \(y^2 = 8x\). Let \(L_1\) and \(L_2\) be the lengths of the common chords of these curves. Then which of the following is true?</p><p>(Based on the figure and solution: find the ratio \(L_1/L_2\).)</p>
Step-by-Step Solution
Key Concept: Find intersection points of circle x² + y² = 9 and parabola y² = 8x by substitution, then calculate the chord length using the distance formula between the two intersection points.
<p><strong>Step 1:</strong> Find intersection points. Substitute y² = 8x into x² + y² = 9:</p><p>x² + 8x = 9 → x² + 8x - 9 = 0 → (x + 9)(x - 1) = 0</p><p>So x = 1 (valid, since x ≥ 0 for parabola) or x = -9 (invalid)</p><p><strong>Step 2:</strong> At x = 1: y² = 8(1) = 8 → y = ±2√2</p><p>Intersection points: (1, 2√2) and (1, -2√2)</p><p><strong>Step 3:</strong> The common chord is vertical (at x = 1) connecting these two points.</p><p>Length L = |2√2 - (-2√2)| = 4√2</p><p><strong>Step 4:</strong> Since there is only ONE common chord of length 4√2, if the question asks for L₁/L₂ ratio where both chords exist, verify the problem statement. The chord length is <strong>4√2 ≈ 5.66</strong></p><p>∴ Answer: A (Common chord length = 4√2)</p>
Correct Answer: A