Chapter 9.6, Problem 75AYU

To determine

To find: The radius and centre of each sphere.

Answer to Problem 75AYU

Solution:

Radius: $\frac{3}{\sqrt{2}}$; Centre: $\left(2,0, -1\right)$

Explanation of Solution

Given:

$2x^2 + 2y^2 + 2z^2 - 8x + 4z = - 1$

Calculation:

The equation of the sphere is:

$\begin{array}{l}2x^2 + 2y^2 + 2z^2 - 8x + 4z = -1\\ x^2 + y^2 + z^2 - 4x + 2z = -\frac{1}{2}\\ \left(x^2 - 4x\right) + y^2 + \left(z^2 + 2z\right) = -\frac{1}{2}\\ \left(x^2 - 4x + 4\right) + y^2 + \left(z^2 + 2z + 1\right) = -\frac{1}{2} + 5\\ {\left(x - 2\right)}^2 + {\left(y - 0\right)}^2 + {\left(z + 1\right)}^2 = \frac{9}{2}\\ {\left(x - 2\right)}^2 + {\left(y - 0\right)}^2 + {\left(z + 1\right)}^2 = {\left(\frac{3}{\sqrt{2}}\right)}^2\end{array}$

Therefore the radius of the sphere is $\frac{3}{\sqrt{2}}$ and the centre is $\left(2,0, -1\right)$.