The question seems to be incomplete. I assume that we have the equation of a sphere, and we want to define equations of the intersection between that sphere and some given planes.
A general sphere of radius R centered at the point (a, b, c) is written as:
(x - a)^2 + (y - b)^2 + (z - c)^2 = R^2
Each of the planes will have a fixed value, to find the intersection, we just need to "fix" that fixed value in the equation above.
a) Intersection with xy-plane
The xy-plane has a z-value of 0 (the x and y values are free) so to write the intersection we only need to fixate z = 0.
Using the general sphere we get:
(x - a)^2 + (y - b)^2 + (0 - c)^2 = R^2
(x - a)^2 + (y - b)^2 + c^2 = R^2
(x - a)^2 + (y - b)^2 = R^2 - c^2
This is a circle, centered at (a, b) in the xy-plane, and with a radius of:
√(R^2 - c^2)
Similarly:
b) intersection with xz-plane
Here we have y = 0, then the equation becomes:
(x - a)^2 + (0 - b)^2 + (z - c)^2 = R^2
(x - a)^2 + b^2 + (z - c)^2 = R^2
(x - a)^2 + (z - c)^2 = R^2 - b^2
c) intersection with yz-plane
Here we have x = 0
(0 - a)^2 + (y - b)^2 + (z - c)^2 = R^2
a^2 + (y - b)^2 + (z - c)^2 = R^2
(y - b)^2 + (z - c)^2 = R^2 - a^2
Finally, in which cases we would write DNE?
Only if the expression at the right (the radius of the circle) are smaller than zero.
For example, if R^2 - a^2 < 0, then the sphere does not intersect the yz-plane.
If you want to learn more you can read:
https://brainly.com/question/9462569
