# Intersection of surface and plane.

• Jimmy5050
In summary, the problem involves finding the vector function of the curve created by the intersection of the plane y=5/2 and the elliptic cylinder (x^2)/4 + (z^2)/6 = 5. The correct parametrization for the ellipse formed is x = sqrt(20)*cos(t) and z = sqrt(30)*sin(t).
Jimmy5050
Parameterizing vector function for intersection of cylinder and plane

## Homework Statement

Problem asks us to find the vector function of the curve which is created when the plane y= 5/2 intersects the ellptic cyl. (x^2)/4 + (z^2)/6 = 5

## The Attempt at a Solution

I know its going to be an ellipse formed...

I took the given ellptic cyl. equation, and divided by 5 to get (x^2)/20 + (z^2)/30 = 1.

***I parameterized by using x=cos(t) and z=sin(t) and got ((cost)^2)/20 + ((sint)^2)/30 =1.

Now, by looking at examples that are somewhat similar, I could tell the answer by looking at number relationships. However, I am unsure of the true way to go about getting my final solution.

My "made up way" of solving was to set either x or z to zero before parameterizing.

My final answers are x=(sqrt[20])cos(t) y=5/2 z=(sqrt[30])sin(t)

I checked my answer using a graphing program, and it is correct, but I am just unsure about going about the TRUE way of solving once I get to the part labeled *** above.

Thanks.

Last edited:
Any takers? I'm sure I have the answer right, just not sure of the "correct" last couple steps to get to that answer so that I can show work properly.

Jimmy5050 said:
***I parameterized by using x=cos(t) and z=sin(t) and got ((cost)^2)/20 + ((sint)^2)/30 =1.
This step seems flaky to me.

The correct parametrization is, I believe, x = a*cos(t), z = b*sin(t). Then you have
$$\frac{a^2 cos^2(t)}{20} + \frac{b^2 sin^2(t)}{30} = 1$$

For this equation to be identically true for all t, it must be that a2 = 20 and b2 = 30, which makes the parametric form of the ellipse (in the x-z plane)
x = $\sqrt{20}$ cos(t)
z = $\sqrt{30}$ sin(t)

I think this is the right way to go about it.

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

The intersection of a surface and a plane is the set of points where the surface and plane meet. This can result in a line, a point, or no intersection at all.

## 2. How do you determine if a surface and plane intersect?

To determine if a surface and plane intersect, you can set up an equation using the equations of the surface and the plane. If the equation has a solution, then the two intersect.

## 3. What are the different types of intersections between a surface and a plane?

The different types of intersections between a surface and a plane are a line, a point, or no intersection. If the equations of the surface and plane are the same, the intersection will be the entire surface. If the equations are parallel, there will be no intersection.

## 4. Can a surface and a plane intersect at more than one point?

Yes, a surface and a plane can intersect at more than one point. This can happen if the surface and plane equations result in multiple solutions when solved together.

## 5. How is the intersection of a surface and a plane useful in mathematics and science?

The intersection of a surface and a plane is useful in mathematics and science as it allows for the visualization and analysis of complex shapes and surfaces. It can also help in solving equations and determining the relationship between different objects or systems.

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