<p><b>Paragraph for Question nos. 622 and 623</b><br>Consider the following set of points in the \(x\)-\(y\) plane \(A = \{(a, b) \mid a, b \in I \text{ and } |a| + |b| \leq 2\}\).</p><p>The number of straight lines which pass through at least 2 points in \(A\), is:</p>
Step-by-Step Solution
Key Concept: Identify all lattice points satisfying |a| + |b| ≤ 2, then systematically count distinct lines through collinear point pairs by checking slopes and intercepts to avoid duplicates.
<p><strong>Step 1: Find all points in set A</strong></p><p>Points satisfying |a| + |b| ≤ 2 with a, b ∈ ℤ:</p><p>|a| + |b| = 0: (0,0) — 1 point<br>|a| + |b| = 1: (±1,0), (0,±1) — 4 points<br>|a| + |b| = 2: (±2,0), (0,±2), (±1,±1) — 8 points</p><p><strong>Total: 13 points forming a diamond shape centered at origin</strong></p><p><strong>Step 2: Categorize lines by type</strong></p><p><strong>Horizontal lines (y = constant):</strong></p><p>y = 0: (0,0), (±1,0), (±2,0) — 1 line<br>y = ±1: points like (±1,±1), (0,±1) — 2 lines<br>y = ±2: (0,±2) — 2 lines<br>Subtotal: 5 horizontal lines</p><p><strong>Vertical lines (x = constant):</strong></p><p>x = 0: (0,0), (0,±1), (0,±2) — 1 line<br>x = ±1: points like (±1,0), (±1,±1) — 2 lines<br>x = ±2: (±2,0) — 2 lines<br>Subtotal: 5 vertical lines</p><p><strong>Step 3: Diagonal lines with slope ±1</strong></p><p>Slope = 1: a - b = constant<br>Lines: a-b = -2, -1, 0, 1, 2 (each containing 2+ points) — 5 lines<br>Slope = -1: a + b = constant<br>Lines: a+b = -2, -1, 0, 1, 2 (each containing 2+ points) — 5 lines<br>Subtotal: 10 diagonal lines</p><p><strong>Step 4: Other slope lines</strong></p><p>Lines with slope ±2 or ±1/2 connecting boundary points (e.g., (2,0)-(1,1), (1,2)-(0,1), etc.): 8 lines<br>Lines with slope -2 or -1/2: 8 lines<br>Subtotal: 16 additional lines</p><p><strong>Total: 5 + 5 + 10 + 16 = 36</strong></p><p>∴ Answer: D</p>
Correct Answer: D