What is the center of the sphere passing thro | vedclass

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(A) Let the equation of the sphere be $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$.Since it passes through the origin $(0, 0, 0)$,we have $d = 0$.Since

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$\left( \frac{1}{2}, 1, 2 \right)$

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(A) Let the equation of the sphere be $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$.Since it passes through the origin $(0, 0, 0)$,we have $d = 0$.Since it passes through $(0, 2, 0)$,we have $0^2 + 2^2 + 0^2 + 2u(0) + 2v(2) + 2w(0) + 0 = 0$,which gives $4 + 4v = 0$,so $v = -1$.Since it passes through $(1, 0, 0)$,we have $1^2 + 0^2 + 0^2 + 2u(1) + 2v(0) + 2w(0) + 0 = 0$,which gives $1 + 2u = 0$,so $u = -1/2$.Since it passes through $(0, 0, 4)$,we have $0^2 + 0^2 + 4^2 + 2u(0) + 2v(0) + 2w(4) + 0 = 0$,which gives $16 + 8w = 0$,so $w = -2$.The center of the sphere is $(-u, -v, -w) = \left( -(-1/2), -(-1), -(-2) \right) = \left( \frac{1}{2}, 1, 2 \right)$.

What is the center of the sphere passing through the four points $(0, 0, 0), (0, 2, 0), (1, 0, 0),$ and $(0, 0, 4)$?

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