<p>A point P(<em>x</em>, <em>y</em>) is called a lattice point if <em>x</em>, <em>y</em> ∈ I (set of integers). Then the total number of lattice points in the interior of the circle \(x^2 + y^2 = a^2\), \(a \neq 0\) cannot be</p>
Step-by-Step Solution
Key Concept: A lattice point (x,y) is strictly inside the circle if x² + y² < a². We need to find which count is impossible by analyzing the symmetry properties of lattice points around the origin and the constraint that a² must allow integer solutions.
<p><strong>Step 1:</strong> Identify which lattice points lie strictly inside: x² + y² < a²</p><p><strong>Step 2:</strong> Observe the 4-fold rotational symmetry. If (x,y) is inside, then (−x,y), (x,−y), (−x,−y) are also inside (for x,y ≠ 0).</p><p><strong>Step 3:</strong> Points on the axes: (±k, 0) and (0, ±k) come in groups of 4 (except origin).</p><p><strong>Step 4:</strong> The origin (0,0) is always inside when a ≠ 0.</p><p><strong>Step 5:</strong> Total count = 1 (origin) + 4m (axis points) + 4n (off-axis symmetric points) = 4(m+n) + 1</p><p><strong>Step 6:</strong> Therefore, the total must be of the form <strong>4k + 1</strong> where k ≥ 0.</p><p><strong>Step 7:</strong> Any number NOT of the form 4k+1 (i.e., numbers ≡ 0, 2, 3 mod 4) cannot be the count of interior lattice points.</p><p>∴ Answer: C (Check which option is not ≡ 1 mod 4)</p>
Correct Answer: C