. Consider the equationx2 + y2 + z2 + Gx + Hy + I z + J = 0and let K = G2 + H2 + I 2 4

Chapter 11, Problem 52

(choose chapter or problem)

Consider the equation

$x^{2}+y^{2}+z^{2}+G x+H y+l z+J=0$

and let $\mathrm{K}=\mathrm{G}^{2}+\mathrm{H}^{2}+\mathrm{I}^{2}-4 \mathrm{J}$.

(a) Prove that the equation represents a sphere if $\mathrm{K}>0$, a point if $K=0$, and has no graph if$\mathrm{K}<0$.

(b) In the case where $\mathrm{K}>0$, find the center and radius of the sphere.

Equation Transcription:

${x}^{2}+{y}^{2}+{z}^{2}+Gx+Hy+Iz+J=0$

$K={G}^{2}+{H}^{2}+{I}^{2}-4J$

K > 0

K = 0

K < 0

Text Transcription:

x^2 + y^2 + z^2 + Gx + Hy + Iz + J = 0

K = G^2 + H^2 + I^2 − 4J

K > 0

K = 0

K < 0

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