<p>In each of the following one or more options are correct. Choose the correct option(s).</p><p>(e) The number of integers \((a \pm n, a \in \mathbb{N}\)) lying inside the region bounded by the circles \(x^2 + y^2 - 2x - 1 = 0\) and \(x^2 + y^2 - 2x - 17 = 0\) is</p>
Step-by-Step Solution
Key Concept: Rewrite both circles in standard form to find their centers and radii, then identify integer lattice points in the annular region between them by checking distance from center (1,0).
<p><strong>Step 1: Convert circles to standard form</strong></p><p>Circle 1: x² + y² - 2x - 1 = 0 → (x-1)² + y² = 2</p><p>Circle 2: x² + y² - 2x - 17 = 0 → (x-1)² + y² = 18</p><p><strong>Step 2: Identify geometry</strong></p><p>Both circles are concentric with center C(1, 0).</p><p>Circle 1: radius r₁ = √2 ≈ 1.414</p><p>Circle 2: radius r₂ = √18 = 3√2 ≈ 4.243</p><p><strong>Step 3: Find integer points in annular region</strong></p><p>Need lattice points (m, n) where: 2 < (m-1)² + n² < 18</p><p>Check systematically:</p><p>• m = 0: n² ∈ (1, 17) → n ∈ {±2, ±3, ±4} = 6 points</p><p>• m = 1: n² ∈ (2, 18) → n ∈ {±2, ±3, ±4} = 6 points</p><p>• m = 2: (n)² ∈ (2, 18) → n ∈ {±2, ±3, ±4} = 6 points</p><p>• m = 3: (2)² + n² ∈ (2, 18) → n² ∈ (-2, 14) → n ∈ {±1, ±2, ±3} = 6 points</p><p>• m = 4: (3)² + n² ∈ (2, 18) → n² ∈ (-7, 9) → n ∈ {±1, ±2} = 4 points</p><p>• m = -1: Same as m = 3 → 6 points</p><p>• m = -2: Same as m = 4 → 4 points</p><p><strong>Step 4: Total count</strong></p><p>Total = 6 + 6 + 6 + 6 + 4 + 6 + 4 = 38 integer points</p><p>∴ Answer: A (38 points)</p>
Correct Answer: A