<p>The number of integral values of \(\alpha\) for which the point \((\alpha - 1,\ \alpha + 1)\) lies in the larger segment of the circle \(x^2 + y^2 - x - y - 6 = 0\) made by the chord whose equation is \(x + y - 2 = 0\) is</p>
Step-by-Step Solution
Key Concept: A point lies in the larger segment if it satisfies the circle inequality AND lies on the correct side of the chord. The larger segment is determined by checking which side of the chord contains the center.
<p><strong>Step 1:</strong> Rewrite the circle equation in standard form.</p><p>x² + y² - x - y - 6 = 0</p><p>(x - 1/2)² + (y - 1/2)² = 6 + 1/4 + 1/4 = 13/2</p><p>Center: C(1/2, 1/2), Radius: r = √(13/2)</p><p><strong>Step 2:</strong> Check which side of the chord x + y - 2 = 0 contains the center.</p><p>At C(1/2, 1/2): 1/2 + 1/2 - 2 = -1 < 0</p><p>The larger segment is where x + y - 2 < 0.</p><p><strong>Step 3:</strong> Apply conditions for point P(α - 1, α + 1) to lie in the larger segment.</p><p><strong>Condition 1 (inside circle):</strong> (α - 1)² + (α + 1)² - (α - 1) - (α + 1) - 6 < 0</p><p>α² - 2α + 1 + α² + 2α + 1 - α + 1 - α - 1 - 6 < 0</p><p>2α² - 2α - 6 < 0</p><p>α² - α - 3 < 0</p><p>Using quadratic formula: α = (1 ± √13)/2 ≈ (1 ± 3.606)/2</p><p>α ∈ (-1.303, 2.303)</p><p><strong>Condition 2 (on larger segment side):</strong> (α - 1) + (α + 1) - 2 < 0</p><p>2α - 2 < 0</p><p>α < 1</p><p><strong>Step 4:</strong> Find intersection of both conditions.</p><p>α ∈ (-1.303, 2.303) ∩ (-∞, 1) = (-1.303, 1)</p><p><strong>Step 5:</strong> Count integral values.</p><p>Integral values: α ∈ {-1, 0}</p><p>∴ Answer: <strong>2</strong></p>
Correct Answer: 2