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.

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.