Intersection of a paraboloid and a plane

In summary, the conversation discusses finding the intersection of the paraboloid z = x^2 + y^2 with the plane x - 2y = 0. The solution involves parametrizing the curve of intersection by setting y = t and finding the corresponding values for x and z. The conversation also mentions a similar question involving more variables and discusses the challenges in finding the intersection using the same method. There is a mention of plotting the equations and verifying the solution.
  • #1
Stevecgz
68
0
Question: Consider the intersection of the paraboloid [tex]z = x^2 + y^2[/tex] with the plane [tex]x - 2y = 0[/tex]. Find a parametrization of the curve of intersection and verify that it lies in each surface.

How I went about it:

[tex]x = 2y[/tex]
[tex]z = (2y)^2 + y^2 = 5y^2[/tex]

Set [tex]y = t[/tex], then

[tex]x = 2t[/tex]
[tex]y = t[/tex]
[tex]z = 5t^2[/tex]

I don't know that my answer is wrong, I'm just not certain if I am going about it the correct way. If someone could let me know I would appreciate it. Thanks.

Steve
 
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  • #2
I plotted both the parabolaoid and the plane in one plot and the parametric spacecurve in another plot (using Maple) and they do appear to agree, also your work is without flaw. Good job.
 
  • #3
Yes, it really is that simple!
 
  • #4
Thanks for the replys.

Steve
 
  • #5
HallsofIvy said:
Yes, it really is that simple!

If only the rest of calculus were this easy :smile:
 
  • #6
A similar question - but more variables are present:

Find the intersection of the plane 2x-y-3z=15 and the paraboloid 3z=(x^2)/16 + (y^2)/9

I was unable to use the above method because of the "z" variable in the plane equation. Can I still use the above method? Or is something else necessary?

I plugged 3z=15+2x-y into the equation for the paraboloid and started getting imaginary numbers, was that correct?

"At the end of the number line is a rainbow made out of nothing but primes..."
 
Last edited:

1. What is the intersection of a paraboloid and a plane?

The intersection of a paraboloid and a plane is the set of points where the paraboloid and the plane meet or intersect.

2. How can I determine the intersection of a paraboloid and a plane?

The intersection of a paraboloid and a plane can be determined by setting the equations of the paraboloid and the plane equal to each other and solving for the variables.

3. What does the intersection of a paraboloid and a plane look like?

The intersection of a paraboloid and a plane can take on different forms depending on the orientation and position of the two shapes. It can be a single point, a line, or a curved shape.

4. Is it possible for a paraboloid and a plane to not intersect?

Yes, it is possible for a paraboloid and a plane to not intersect. This can occur if the two shapes are parallel or if they are positioned in such a way that they do not intersect.

5. What are the real-world applications of understanding the intersection of a paraboloid and a plane?

The intersection of a paraboloid and a plane is commonly used in engineering and architecture to design and create curved structures such as domes, arches, and bridges. It is also used in physics and optics to study the behavior of light and sound waves as they intersect with curved surfaces.

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