# Intersection of a sphere and plane

• -EquinoX-
In summary, the sphere has the equation: (x-4)^2 + (y-3)^2 + (z+2)^2 = 20. If you are in the xy plane, then z=0. What's wrong with setting z=0? Then if I am asked to find the yz plane I set x to 0? the thing is then all intersection type will be a circle.. and I am not sure if that's the right answer. The radius of the sphere is sqrt(20). The center of the sphere is (4,3,-2). The farthest coordinate plane is the yz plane. The center of the sphere is 4 units away from that. sqrt(20
-EquinoX-

## Homework Statement

Say I have a sphere with the equation:

(x-4)^2 + (y-3)^2 + (z+2)^2 = 20

How do I find the intersection with the xy-plane? Is the intersection at a point or it forms a circle

## The Attempt at a Solution

Hmm.. I tried to set z equal to 0 to solve this problem, however I think that's not the way to solve this problem

If you are in the xy plane, then z=0. What's wrong with setting z=0?

then if I am asked to find the yz plane I set x to 0? the thing is then all intersection type will be a circle.. and I am not sure if that's the right answer

The radius of the sphere is sqrt(20). The center of the sphere is (4,3,-2). The farthest coordinate plane is the yz plane. The center of the sphere is 4 units away from that. sqrt(20)>4. They are all circles.

ok, so what if the question is

"How does the sphere intersect each of the x-axis, y-axis, z-axis"?

all of them? how do I find the equation though?

-EquinoX- said:
all of them? how do I find the equation though?

I retracted that quick (and wrong) answer. For the x-axis you have to figure out if the circle in the x-y plane intersects the x-axis. More simply, where you hit the x-axis is where y=0, and z=0. Etc.

if that's the case then what's left is just (x-4)^2 = 20, correct? in that case it will intersect at two points as it's a quadratic equation

No, what's left if y=0 and z=0 is (x-4)^2+3^2+2^2=20.

yes that's true, either way all will be in that form and solving it is just basically a quadratic equation and we get two points for each plane (xy-plane, yz-plane, xz-plane), correct?

No. That's why I retracted my answer. The intersections with the z-axis have to satisfy (z+2)^2+3^2+4^2=20. Now what?

z^2+4z+9 to make it more simple, right... and that is a quadratic equation, so why can't we get two points out of it?

(z+2)^2=(-5) to put it even more simply. Some quadratic equation don't have any real solutions. Face it.

oh, you're right.. we can't take the square root of 5... so for a special case for the z-axis, it doesn't intersect anywhere, not even at a point.. right?

Right. You meant sqrt(-5), right?

the equation with the intersection at the y-axis is simply:
= (y-3)^2 = 0
= y-3 = 0
= y = 3

so it just intersects at a point

Last edited:
-EquinoX- said:
the equation with the intersection at the y-axis is simply:
= (y-3)^2 = 0
= y-3 = 0
= y = 3

so it just intersects at a point

Sure.

## What is the intersection of a sphere and a plane?

The intersection of a sphere and a plane is the set of points where the sphere and the plane intersect or touch each other.

## What are the different possible outcomes of the intersection of a sphere and a plane?

The intersection of a sphere and a plane can result in a circle, a point, or no intersection at all.

## How does the position of the plane relative to the sphere affect the intersection?

If the plane is tangent to the sphere, the intersection will be a circle. If the plane is parallel to the sphere, there will be no intersection. And if the plane intersects the sphere at an angle, the intersection will be a point.

## What is the equation for calculating the intersection of a sphere and a plane?

The equation for calculating the intersection of a sphere and a plane is (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2, where (a, b, c) is the center of the sphere and r is its radius.

## How is the intersection of a sphere and a plane used in real life?

The intersection of a sphere and a plane is used in many different areas of science and engineering, such as computer graphics, architecture, and physics. It can be used to create 3D models, calculate the trajectory of a projectile, or determine the shape of a planet.

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