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

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

2020-12-26
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.
$$(x - h)^{2} + (y - k)^{2} + (z - l)^{2} = r^{2} (1)$$
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.
$$(x^{2} + 8x + 4^{2} - 4^{2}) + (y^{2} - 6y + 3^{2} - 3^{2}) + (z^{2} + 2z + 1^{2} - 1^{2}) + 17 = 0$$
$$(x^{2} + 8x +4^{2}) + (y^{2} - 6y + 3^{2}) + (z^{2} + 2z + 1^{20} + (-16 - 9 - 1) + 17 = 0$$
$$(x + 4)^{2} + (y - 3)^{2} + (z + 1)^{2} = 26 - 17$$
$$(x + 4)^{2} + (y - 3)^{2} + (z + 1)^{2} = 9$$
$$[x - (-4)]^{2} + (y - 3)^{2} + [z - (-1)]^{2} = (3)^{2} (2)$$
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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