Finding the equation of a hyperboloid

  • Thread starter RKOwens4
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In summary, the conversation discusses finding the equation of a hyperboloid of one sheet passing through specific points. The formula for a hyperboloid of one sheet is given, and the conversation focuses on determining the value of c in the equation. The solution is found by substituting the given points and solving for c.
  • #1
RKOwens4
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Homework Statement



"Find the equation of the hyperboloid of one sheet passing through the points (+-2, 0, 0), (0, +-4, 0) and (+-4, 0, 7), (0, +-8, 7)."

(What I mean by "+-" is the plus sign with the minus sign below it, read "plus or minus".)

Homework Equations



Equation for a hyperboloid of one sheet: (x/a)^2 + (y/b)^2 - (z/c)^2 = 1.

The Attempt at a Solution



I'm able to get the first part of the equation figured out easily. I get (x/2)^2 + (y/4)^2 - (z/?)^2 = 1. But I can't figure out what to put for the denominator in the z part. I thought maybe square root of 7, but that's wrong. I also tried 7, but that's incorrect. I know this is a really minor thing to be posting a whole thread about, but I can't figure it out and if anyone could help, it'd be appreciated.
 
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  • #2
Use

[tex] (\frac{x}{2})^{2} + (\frac{y}{4})^{2} - (\frac{z}{c})^{2} = 1 [/tex]

and simply subsistute your points where,

[tex] Z \neq 0 [/tex]

and from there you should be able to compute c.

[tex] c = \pm \frac{7}{\sqrt{3}} \approx \pm 4.041[/tex]
 
Last edited:
  • #3
That's correct. Thanks!
 

1. What is a hyperboloid?

A hyperboloid is a three-dimensional geometric shape that can be formed by rotating a hyperbola around its axis.

2. How do you find the equation of a hyperboloid?

The equation of a hyperboloid can be found by using the standard form equation:
(x-h)^2/a^2 + (y-k)^2/b^2 - (z-z0)^2/c^2 = 1
where (h,k,z0) is the center of the hyperboloid and a, b, and c are the distances from the center to the vertices along the x, y, and z axes respectively.

3. What are the different types of hyperboloids?

There are two types of hyperboloids:
1. One-sheet hyperboloids, which have one continuous surface without any boundary or edges.
2. Two-sheet hyperboloids, which have two continuous surfaces that are mirror images of each other and are separated by a boundary or ring.

4. Can the equation of a hyperboloid be written in other forms?

Yes, the equation of a hyperboloid can also be written in a general form as:
Ax^2 + By^2 + Cz^2 + Dx + Ey + Fz + G = 0
or in a parametric form as:
x = a cosh(u) cos(v), y = b cosh(u) sin(v), z = c sinh(u)
where u and v are parameters and cosh and sinh are hyperbolic functions.

5. What are some real-life applications of hyperboloids?

Hyperboloids have various applications in engineering, architecture, and physics. Some examples include:
1. Cooling towers for power plants
2. Reflectors for radio telescopes
3. Antenna structures
4. Bridges and tunnels
5. Hyperboloid structures in architecture such as the Guggenheim Museum in Bilbao, Spain.

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