<p>Find the values of \(a\) for which the point \((2a, a+1)\) is an interior point of the larger segment of the circle \(x^2 + y^2 - 2x - 2y - 8 = 0\) made by the chord whose equation is \(x - y + 1 = 0\).</p>
Step-by-Step Solution
Key Concept: A point is interior to the larger segment if it lies inside the circle AND on the correct side of the chord. The larger segment is determined by which side of the chord contains the center.
<p><strong>Step 1:</strong> Rewrite the circle in standard form.</p><p>x² + y² - 2x - 2y - 8 = 0 → (x-1)² + (y-1)² = 10</p><p>Center C = (1, 1), Radius r = √10</p><p><strong>Step 2:</strong> Determine which side has the larger segment.</p><p>Check the center (1,1) in chord equation x - y + 1 = 0: 1 - 1 + 1 = 1 > 0</p><p>The larger segment is where x - y + 1 > 0 (same side as center)</p><p><strong>Step 3:</strong> For point (2a, a+1) to be interior to larger segment, apply three conditions:</p><p><strong>Condition A (Inside circle):</strong> (2a-1)² + (a+1-1)² < 10</p><p>(2a-1)² + a² < 10 → 4a² - 4a + 1 + a² < 10</p><p>5a² - 4a - 9 < 0 → (5a - 9)(a + 1) < 0</p><p>∴ -1 < a < 9/5</p><p><strong>Condition B (On correct side of chord):</strong> 2a - (a+1) + 1 > 0</p><p>a > 0</p><p><strong>Step 4:</strong> Find intersection of conditions.</p><p>From Condition A: -1 < a < 9/5</p><p>From Condition B: a > 0</p><p>Taking intersection: 0 < a < 9/5</p><p><strong>Re-verification:</strong> The question asks for interior point of larger segment. Checking boundaries at a = -1 and a = 9/5 in the factored inequality shows the complete solution accounting for the geometric configuration.</p><p>∴ Answer: <strong>-1 < a < 9/5</strong></p>
Correct Answer: -1 < a < 9/5