Question

Show the equation \(2{x^2} + 2{y^2} + 2{z^2} = 8x - 24z + 1\) as an equation of a sphere and determine the center and radius of the sphere.

Short answer

The equation \(2{x^2} + 2{y^2} + 2{z^2} = 8x - 24z + 1\) represents a sphere. The center of the sphere is \({\rm{(2,0, - 6)}}\)and the radius of the sphere is \(\left( {\frac{9}{{\sqrt 2 }}} \right).\)

Step-by-step solution

Formula of the expression of a sphere.

Write the expression find an equation of the sphere with center\({\rm{C(h, k, l)}}\)and radius\({\rm{r}}{\rm{.}}\)

\({(x - h)^2} + {(y - k)^2} + {(z - l)^2} = {r^2}\) …… (1)

Here,

\({\rm{(h, k, l)}}\)Is the center of the sphere, which is\((3,8,1)\)and

\({\rm{r}}{\rm{.}}\)Is the radius of the sphere.

Use the equation (1) for further calculation.

Rearrange equation\(2{x^2} + 2{y^2} + 2{z^2} = 8x - 24z + 1\) as follows.

\(\begin{array}{l}2\left( {{x^2} - 4x + {2^2} - {2^2}} \right) + 2\left( {{y^2}} \right) + 2\left( {{z^2} + 12z + {6^2} - {6^2}} \right) = 1\\2\left( {{x^2} - 4x + {2^2}} \right) + 2{y^2} + 2\left( {{z^2} + 12z + {6^2}} \right) + ( - 8 - 72) = 1\\2{(x - 2)^2} + 2{(y - 0)^2} + 2{(z + 6)^2} = 81\\{(x - 2)^2} + {(y - 0)^2} + {(z + 6)^2} = \frac{{81}}{2}\\{(x - 2)^2} + {(y - 0)^2} + {(z - ( - 6))^2} = {\left( {\frac{9}{{\sqrt 2 }}} \right)^2}\end{array}\) …… (2)

Equation (2) is similar to equation (1).

Therefore, the equation\(2{x^2} + 2{y^2} + 2{z^2} = 8x - 24z + 1\) represents a sphere.

Compare equation (2) with equation (1).

\(\begin{array}{l}h = 2\\k = 0\\l = - 6\\r = \frac{9}{{\sqrt 2 }}\end{array}\)

Thus, the center of the sphere is \({\rm{(2,0, - 6)}}\)and the radius of the sphere is \(\left( {\frac{9}{{\sqrt 2 }}} \right).\)