<p>A line drawn through the point \(P(4, 7)\) cuts the circle \(x^2 + y^2 = 9\) at the points \(A\) and \(B\), then \(PA \cdot PB\) is equal to</p>
Step-by-Step Solution
Key Concept: Use the Power of a Point theorem: for any line through external point P intersecting a circle at points A and B, the product PA·PB equals the power of point P with respect to the circle, which is |OP|² - r².
<p><strong>Step 1:</strong> Identify the circle: x² + y² = 9 has center O(0, 0) and radius r = 3.</p><p><strong>Step 2:</strong> Check that P(4, 7) is outside the circle: |OP|² = 4² + 7² = 16 + 49 = 65 > 9. ✓</p><p><strong>Step 3:</strong> Apply Power of a Point theorem: For any line through external point P intersecting circle at A and B,</p><p>PA · PB = |OP|² - r² = 65 - 9 = 56</p><p><strong>Step 4:</strong> This result is independent of which line through P is chosen (whether secant or any other configuration through P).</p><p>∴ Answer: PA · PB = <strong>56</strong></p>
Correct Answer: B