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📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Question: Find the point on the line \(y = 2x + 1\) in the plane that is closest to the point \((4, 3)\).

📌 Solution: We minimize the distance from a point \((x, 2x+1)\) on the line to \((4, 3)\). The squared distance is:

📌 D = (x - 4)^2 + (2x + 1 - 3)^2 = (x - 4)^2 + (2x - 2)^2.

📌 D = (x^2 - 8x + 16) + (4x^2 - 8x + 4) = 5x^2 - 16x + 20.

📌 rac{dD}{dx} = 10x - 16 = 0 \Rightarrow x = rac{8}{5}.

📌 Then $y = 2\left(rac{8}{5}

📌 ight) + 1 = rac{16}{5} + rac{5}{5} = rac{21}{5}$.

📌 Thus, the closest point is $\left(rac{8}{5}, rac{21}{5}

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