# Show that the equation represents a sphere, and find its center and radius. x^{2} + y^{2} + z^{2} + 8x - 6y + 2z + 17 =0

Show that the equation represents a sphere, and find its center and radius.
${x}^{2}+{y}^{2}+{z}^{2}+8x-6y+2z+17=0$
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Mayme
Consider a sphere with center C(h, k, l) and radius r.
Formula:
Write the expression to find an equation of a sphere with center C (h, k, l) and radius r.
$\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}+\left(z-l{\right)}^{2}={r}^{2}\left(1\right)$
Here,
(h, k, l) is the center of a sphere and
r is the radius of a sphere.
Rearrange the expression ${x}^{2}+{y}^{2}+{z}^{2}+8x-6y+2z+17=0$ as follows.
$\left({x}^{2}+8x+{4}^{2}-{4}^{2}\right)+\left({y}^{2}-6y+{3}^{2}-{3}^{2}\right)+\left({z}^{2}+2z+{1}^{2}-{1}^{2}\right)+17=0$
$\left({x}^{2}+8x+{4}^{2}\right)+\left({y}^{2}-6y+{3}^{2}\right)+\left({z}^{2}+2z+{1}^{20}+\left(-16-9-1\right)+17=0$
$\left(x+4{\right)}^{2}+\left(y-3{\right)}^{2}+\left(z+1{\right)}^{2}=26-17$
$\left(x+4{\right)}^{2}+\left(y-3{\right)}^{2}+\left(z+1{\right)}^{2}=9$
$\left[x-\left(-4\right){\right]}^{2}+\left(y-3{\right)}^{2}+\left[z-\left(-1\right){\right]}^{2}=\left(3{\right)}^{2}\left(2\right)$
Equation (2) is similar to equation (1).
Therefore, the equation $x62+{y}^{2}+{z}^{2}+8x-6y+2z+17=0$ represents a sphere.
Compare equation (2) with equation (1).
$h=-4$
$k=3$
$l=-1$
$r=3$
Thus, the center of the spere is (-4, 3, -1) and the radius of the sphere is 3.
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