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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 This is the equation of a **torus** centered along the \(z\)-axis with major radius \(3c\) and minor radius \(3c\), but more precisely, from the original form \(

📌 ho = 3c \sin(\phi)\), this describes a **sphere** of radius \(rac{3c}{2}\) centered at \((0, 0, rac{3c}{2})\) in Cartesian coordinates. To verify, note that:

📌 ho = 2a \sin\phi \Rightarrow ext{sphere of radius } a ext{ centered at } (0,0,a).

📌 Here, \(2a = 3c \Rightarrow a = rac{3c}{2}\), so the shape is a sphere of radius \(rac{3c}{2}\) centered at \((0, 0, rac{3c}{2})\).

📌 Thus, the equation describes a sphere of radius \(rac{3c}{2}\) centered at \((0, 0, rac{3c}{2})\).

📌 oxed{ ext{A sphere of radius } \dfrac{3c}{2} ext{ centered at } \left(0, 0, \dfrac{3c}{2}

📌 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:

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