<h2><u>
Answer</u>
:</h2>
(a) x² + y² + z² + 18(x - y + z) + 218 = 0
(b) (x + 9)² + (y - 9)² + 56 = 0
<h2><u>
Step-by-step explanation:</u></h2>
<em>The general equation of a sphere of radius r and centered at C = (x₀, y₀, z₀) is given by;</em>
(x - x₀)² + (y - y₀)² + (z - z₀)² = r² ------------------(i)
<em>From the question:</em>
The sphere is centered at C = (x₀, y₀, z₀) = (-9, 9, -9) and has a radius r = 5.
<em>Therefore, to get the equation of the sphere, substitute these values into equation (i) as follows;</em>
(x - (-9))² + (y - 9)² + (z - (-9))² = 5²
(x + 9)² + (y - 9)² + (z + 9)² = 25 ------------------(ii)
<em>Open the brackets and have the following:</em>
(x + 9)² + (y - 9)² + (z + 9)² = 25
(x² + 18x + 81) + (y² - 18y + 81) + (z² + 18z + 81) = 25
x² + 18x + 81 + y² - 18y + 81 + z² + 18z + 81 = 25
x² + y² + z² + 18(x - y + z) + 243 = 25
x² + y² + z² + 18(x - y + z) + 218 = 0 [<em>equation has already been normalized since the coefficient of x² is 1</em>]
<em>Therefore, the equation of the sphere centered at (-9,9, -9) with radius 5 is:</em>
x² + y² + z² + 18(x - y + z) + 218 = 0
(2) To get the equation when the sphere intersects a plane z = 0, we substitute z = 0 in equation (ii) as follows;
(x + 9)² + (y - 9)² + (0 + 9)² = 25
(x + 9)² + (y - 9)² + (9)² = 25
(x + 9)² + (y - 9)² + 81 = 25 [<em>subtract 25 from both sides</em>]
(x + 9)² + (y - 9)² + 81 - 25 = 25 - 25
(x + 9)² + (y - 9)² + 56 = 0
The equation is therefore, (x + 9)² + (y - 9)² + 56 = 0
