Answer
See the explanation
Work Step by Step
To find equations for spheres in space, we need to know the center coordinates and the radius of the sphere. The general equation for a sphere in three-dimensional space is:
$(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$
where $(h, k, l)$ represents the center coordinates of the sphere, and r represents the radius.
Let's consider a few examples:
Example i:
Center:$ (2, -3, 1)$
Radius: $4$
The equation for this sphere would be:
$(x - 2)^2 + (y + 3)^2 + (z - 1)^2 = 16$
Example ii:
Center: $(0, 0, 0)$
Radius: $5$
The equation for this sphere would be:
$x^2 + y^2 + z^2 = 25$
Example iii:
Center: $(-1, 2, -4)$
Radius: $3$
The equation for this sphere would be:
$(x + 1)^2 + (y - 2)^2 + (z + 4)^2 = 9$
In each example, the equation represents all the points in space that are equidistant from the center of the sphere.
