A sphere is given by the equation \(x^2+y^2+z^2-10z=0\). If one endpoint of a diameter is \((-3,4,5)\), then what are the coordinates of the opposite endpoint?

\((-3,-4,-5)\)
\((3,4,5)\)
\((3,4,-5)\)
\((-3,4,-5)\)

Correct answer: \((3,4,5)\)

Solution

The sphere \(x^2+y^2+z^2-10z=0\) has center \((0,0,5)\). Since one endpoint of a diameter is \((-3,4,5)\), the opposite endpoint is its reflection across the center, which is \((3,4,5)\).

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