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Hour 1: 150 - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question:** How many lattice points lie on the ellipse \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \)?

📌 Solution:** We seek integer solutions \( (x, y) \) to \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \), or equivalently:

📌 9x^2 + 25y^2 = 225

📌 We test integer values of \( x \) from \( -5 \) to \( 5 \), since \( x^2 \leq 25 \Rightarrow |x| \leq 5 \).

📌 For each \( x \), check if \( 225 - 9x^2 \) is divisible by 25 and the result for \( y^2 \) is a perfect square.

📌 \( x = 0 \): \( 9(0) + 25y^2 = 225 \Rightarrow y^2 = 9 \Rightarrow y = \pm 3 \) → 2 points

📌 \( x = \pm 1 \): \( 9(1) = 9 \Rightarrow 25y^2 = 216 \Rightarrow y^2 = 8.64 \) → not integer

📌 \( x = \pm 2 \): \( 9(4) = 36 \Rightarrow 25y^2 = 189 \Rightarrow y^2 = 7.56 \) → no

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