#### 4e-07**Question:** How many lattice points lie on the ellipse defined by \(4x^2 + 9y^2 = 36\)?
["### How Many Lattice Points Lie on the Ellipse Defined by (4x^2 + 9y^2 = 36)?", "When studying conic sections, lattice points—integer-coordinate points—on geometric curves attract deep interest, especially ellipses defined by symmetric equations. The given ellipse, given by:", "[\n4x^2 + 9y^2 = 36\n]", "presents a classic problem: determining exactly how many lattice points (points ((x, y)) where both (x) and (y) are integers) satisfy this equation.", "#### Step 1: Rewrite the Equation in Standard Form", "Divide both sides of the equation by 36 to convert it to standard ellipse form:", "[\n\frac{x^2}{9} + \frac{y^2}{4} = 1\n]", "This shows the ellipse has semi-major axis (a = 3) (along the (x)-axis) and semi-minor axis (b = 2) (along the (y)-axis).", "#### Step 2: Identify Possible Integer Values for (x) and (y)", "Since (x^2 \leq 9), (x) must satisfy (-3 \leq x \leq 3). Similarly, (y) must satisfy (-2 \leq y \leq 2) due to (y^2 \leq 4). We test all integer values in this range.", "#### Step 3: Check Each Candidate Pair ((x, y))", "We substitute each integer (x) in ([-3, 3]), solve for (y^2), and check whether (y^2) is a perfect square (to ensure (y) is integer).", "[\n4x^2 + 9y^2 = 36 \quad \Rightarrow \quad 9y^2 = 36 - 4x^2 \quad \Rightarrow \quad y^2 = \frac{36 - 4x^2}{9}\n]", "Now evaluate:", "- (x = -3, 3 ):\n (x^2 = 9) → (4x^2 = 36) → (9y^2 = 0) → (y = 0)\n Points: ((3, 0), (-3, 0)) — 2 points", "- (x = -2, 2):\n (x^2 = 4) → (4x^2 = 16) → (9y^2 = 20) → (y^2 = \frac{20}{9}) (not integer) → no lattice points", "- (x = -1, 1):\n (x^2 = 1) → (4x^2 = 4) → (9y^2 = 32) → (y^2 = \frac{32}{9}) (not integer) → no lattice points", "- (x = 0):\n (x^2 = 0) → (9y^2 = 36) → (y^2 = 4) → (y = \pm 2)\n Points: ((0, 2), (0, -2)) — 2 points", "#### Step 4: Summary of Lattice Points", "From all tested values, only four integer pairs satisfy the equation:", "[\n(3, 0),\ (-3, 0),\ (0, 2),\ (0, -2)\n]", "These are the only lattice points on the ellipse.", "#### Conclusion", "The ellipse defined by (4x^2 + 9y^2 = 36) contains exactly 4 lattice points. This straightforward enumeration confirms a key technique in computational geometry: bounding variables via symmetry and checking feasible integer solutions within the curve’s extent.", "#### Expand Your Knowledge\nExplore how adjusting coefficients or shifting axes affects lattice point counts—important in number theory and discrete geometry.", "---", "Tagline: Discover how symmetry and algebra reveal hidden lattice structures on ellipses—drop a comment if you’ve solved similar Diophantine geometry problems!"]
