Answer:
The standard equation of the sphere is 
Step-by-step explanation:
From the question, the end point are (3,-2,4) and (7,12,4)
Since we know the end points of the diameter, we can determine the center (midpoint of the two end points) of the sphere.
The midpoint can be calculated thus
Midpoint = 
Let the first endpoint be represented as  and the second endpoint be
 and the second endpoint be  .
.
Hence, 
Midpoint = 
Midpoint = 
Midpoint = 
Midpoint = 
This is the center of the sphere. 
Now, we will determine the distance (diameter) of the sphere 
The distance is given by






This is the diameter
To find the radius, r 
From 

∴ 
r = 
Now, we can write the standard equation of the sphere since we know the center and the radius
Center of the sphere is 
Radius of the sphere is 
The equation of a sphere of radius r and center  is given by
 is given by

Hence, the equation of the sphere of radius  and center
 and center  is
 is


This is the standard equation of the sphere