How Does a Sphere Intersect Each Coordinate Plane and Axis?

In summary, To find the equation of the shapes where the surface intersects each coordinate plane, set one of the variables in the sphere equation to zero. For the yz-plane, substitute x=0 to get a circle equation. For the z-axis, substitute x=0 and y=0.
  • #1
Theansweris42
1
0
I have a sphere with the equation
(x-3)2+(y-2)2+(z+5)2=36

two question:
1. how do I find the equation of the shapes where the surface intersects each coordinate plane.
2. How do I find the points where the surface intersects each coordinate axis.

I think for 1. that it will be a circle but how do I find the equation of that circle? What part of the sphere equation do I need to set to zero?
 
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  • #2
A point (x,y,z) is in both the sphere and the yz-plane iff:
[tex](x-3)^2 + (y-2)^2 + (z+5)^2 = 36[/tex]
[tex]x=0[/tex]
So just substitute x=0 in the sphere equation to get a circle equation. For the intersection with the z-axis we have:
[tex](x-3)^2 + (y-2)^2 + (z+5)^2 = 36[/tex]
[tex]x=0[/tex]
[tex]y=0[/tex]
 

FAQ: How Does a Sphere Intersect Each Coordinate Plane and Axis?

How can the intersection of a sphere and a plane be calculated?

The intersection of a sphere and a plane can be calculated using the formula for a circle and the equation of a plane. The center point of the circle can be found by setting the equation of the plane equal to the equation of the sphere and solving for the values of x, y, and z. The radius of the circle can then be calculated using the distance formula between the center point and any point on the plane that lies on the circle.

Can a sphere intersect with a plane at more than one point?

Yes, a sphere can intersect with a plane at more than one point. This can occur when the plane passes through the center of the sphere, resulting in a circle of intersection. It can also occur when the plane is tangent to the sphere, resulting in a single point of intersection.

What is the shape of the intersection when a sphere intersects with a plane?

The shape of the intersection between a sphere and a plane can vary depending on the relative positions of the sphere and the plane. It can be a circle, a single point, or in some cases, no intersection at all. It is also possible for the intersection to be an ellipse or a parabola, depending on the orientation of the plane and the size of the sphere.

How does the radius of the sphere affect the intersection with a plane?

The radius of the sphere can greatly affect the intersection with a plane. If the sphere is larger, the intersection will be larger and may be more pronounced. If the sphere is smaller, the intersection may be smaller or even non-existent, if the plane is far enough away from the sphere.

Can a sphere intersect with a plane in three-dimensional space?

Yes, a sphere can intersect with a plane in three-dimensional space. In fact, the intersection of a sphere and a plane is a common problem in 3D geometry and is used in various applications such as computer graphics and engineering. The intersection can occur at any angle and in any orientation, resulting in different shapes and sizes of the intersection.

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