# Intersection of a sphere and plane problem

• prace
In summary, the person is trying to solve for y and z when x=0, but they are not able to do so because the intersection is not a single point.
prace
Hello,

I was wondering if anyone could offer some advice on this one. I have a multi-part problem, in which I can't get the second part. It starts like this:

(a)
Find the equation of the sphere passes through the point (6,-2,3) and has a center of (-1,2,1).

So I did this find and came up with:

(x+1)²+(y-2)²+(z-1)² = 69

(b)
Find the curve in which this sphere intersects the yz-plane.

So this is where I am stuck...

The zy plane is the set $\Pi_{zy}$ of all points (x,y,z ) of R^3 for which x=0.

and your sphere (I haven't checked if you found the correct radius) is the set $\mathbb{S}_{\sqrt{69}}$ of all points (x,y,z) of R^3 that obey the equation (x+1)²+(y-2)²+(z-1)² = 69.

So, if you define the set $\mbox{Intersection}(\Pi_{zy},\mathbb{S}_{\sqrt{69}})$ of as the points (x,y,z) of R^3 that obey both to the x=0 and the (x+1)²+(y-2)²+(z-1)² = 69 condition, then points of this set are both the plane and the sphere. Logical?

Last edited:
Hmm.. This is a little confusing for me. Sorry about that. So... I am not sure what the notation $\Pi_{zy}$ means, but I am guessing that it means the set of points on a sphere in the zy plane? But what you are basically telling me to do, or to think about, is to let x = 0, then I can solve a system of 2 equations and two unknowns to solve for y and z when x = 0. How does that sound?

First, do you understand that the intersection is a curve- one dimensional- and so can't be written as a single equation in 3 dimensions?

Yes, you can let x= 0. But then you can't "solve a system of two equations and two unknowns to solve for y and z when x= 0" because, first, you don't have two equations, and second, the intersection is not a single point!

letting x= 0 you get 1+ (y- 2)2+ (z- 1)2= 69 or
(y- 2)2+ (z- 1)2= 68. That's the equation for a circle. A standard parametrization for a circle is to let $\theta$ be the angle a radius makes with an axis. In particular, what must y and z be, in terms of $\theta$ so that your equation becomes
$68cos^2(\theta)+ 68sin^2(\theta)= 68$?

## 1. What is the intersection of a sphere and plane problem?

The intersection of a sphere and plane problem is a mathematical problem that involves finding the points where a three-dimensional sphere and a two-dimensional plane intersect. This problem is often encountered in geometry and can have multiple solutions depending on the position and orientation of the sphere and plane.

## 2. How do you find the intersection points of a sphere and plane?

To find the intersection points of a sphere and plane, you can use the equations of the sphere and plane in three-dimensional space. The intersection points will be the solutions to these equations, which can be found using algebra or geometric methods. Alternatively, you can also use software or computer programs to calculate the intersection points.

## 3. Can a sphere and plane intersect at more than two points?

Yes, a sphere and plane can intersect at more than two points. The number of intersection points depends on the relative positions and orientations of the sphere and plane. For example, if the plane passes through the center of the sphere, it will intersect at an infinite number of points. However, if the plane is tangent to the sphere, it will only intersect at one point.

## 4. Are there any real-life applications of the sphere and plane intersection problem?

Yes, the sphere and plane intersection problem has many real-life applications, especially in the fields of physics, engineering, and computer graphics. For example, it can be used to calculate the trajectory of a ball bouncing off a flat surface, or to design the shape of a car headlight reflector. It is also essential in computer graphics for creating 3D models and animations.

## 5. Is there a general formula for finding the intersection points of any sphere and plane?

Yes, there is a general formula for finding the intersection points of any sphere and plane, called the quadratic formula. This formula involves the coordinates and radius of the sphere, as well as the equation of the plane. However, depending on the specific scenario, the calculation may become more complex, and alternative methods may need to be used.

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