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\(\boxed{2 \pm \sqrt6}\)1. **Frage: - Coaching Toolbox

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Question:** An entomologist specializing in insect ecology is studying the movement patterns of a certain species of insect, modeled by the equation \((x - 3)^2 + (y - 4)^2 = 25\). Determine the number of integer coordinate points \((x, y)\) that lie on this circle.

📌 The given equation represents a circle centered at \((3, 4)\) with radius 5. The equation is:

📌 To find integer solutions, consider \(x = 3 + a\) and \(y = 4 + b\), where \(a^2 + b^2 = 25\). We need integer pairs \((a, b)\) such that:

📌 a^2 + b^2 = 25

📌 Possible values for \(a^2\) are \(0, 1, 4, 9, 16, 25\). We check each:

📌 \(a^2 = 0\), \(b^2 = 25\) gives \((a, b) = (0, \pm 5)\)

📌 \(a^2 = 1\), \(b^2 = 24\) is not possible

📌 \(a^2 = 4\), \(b^2 = 21\) is not possible

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