Gfind an equation of the sphere with center (3, −10, 4) and radius 5. (x−3)2+(y+10)2+(z−4)2=25 use an equation to describe its intersection with each of the coordinate planes. (if the sphere does not intersect with the plane, enter dne.)

The Equation of the sphere with center (3, -10, 4) and radius 5 is correctly given as:

, 

The x-y axis is the set of all points (ordered triples) of the form (x, y, 0), where x, y can be any point, but the z-coordinate is 0.

the intersection of the sphere and this plane is found by letting z=0:



the last equation is the standard equation of the circle with center (3, -10) and radius 3.

Indeed, we expect the intersection of a sphere with a plane to be a circle.

Similarly the y-z plane is represented by (0, y, z) and the x-z plane by (x, 0, z), 

The intersection of the sphere with the plane y-z is:



and the intersection of the sphere with the x-z plane is :



which makes no sense since the left hand side is positive (or at least 0).. This means that there is no intersection of the sphere with the x-z plane.

 

intersection with the x-y plane: 

intersection with the y-z plane: 

intersection with the x-y plane: none