Find the equation of a sphere.

  • Thread starter bobbarkernar
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In summary: The final answer is what is most important.In summary, we need to find an equation of a sphere given the endpoints of one of its diameters. Using the midpoint formula, we can find the center of the sphere. Then, plugging in the center and the radius (which is found by taking the square root of the sum of the squares of the differences between the endpoints), we can write the equation of the sphere in the form (x-h)^2+(y-k)^2+(z-l)^2=r^2.
  • #1
bobbarkernar
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Homework Statement


find an equation of a sphere if one of its diameters has endpoints (2,1,4)and
(4,3,10)


Homework Equations





The Attempt at a Solution



it should be in the form of (x-h)^2+(y-k)^2+(z-l)^2=r^2
i think i found the radius by:
r= sqrt((4-2)^2+(3-1)^2+(10-4)^2)
r= sqrt(44)/2
r= sqrt(11)
im not sure about the coordinates for the center i think i should just take the difference between the two endpoints of the diameter?
if someone could please give me some info thank you
 
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  • #2
nvm i figured it i need to use midpt formula ty anyway
 
  • #3
Your equation for the radius certainly seems to be fine, but the centre is not calculated by the difference of the points.

Consider something simple like the real line. If you want to find the point between 3 and 1, you take the average of these two points. That is

[tex]\frac{x_1+x_2}{2}[/tex]

[tex]\frac{1+3}{2}=2[/tex]

This applies in higher dimensional space [tex]\mathbf{R^3}[/tex] as well, and can be worked out co-ordinate wise. This will give you the centre of the circle.
 
  • #4
Yes the midpoint formula would have been a lot easier huh? Just sub those points into the spheres equation >.<
 
  • #5
The formula for the radius is not quite "fine"!

You have
r= sqrt((4-2)^2+(3-1)^2+(10-4)^2)
r= sqrt(44)/2
r= sqrt(11)
Your final result is good by that first equation does not have the "/2".
 
  • #6
Yes, I noticed this mistake as well, though I didn't think that it warranted comment. Seeing as how bobbarkernar did indeed derive the final result properly, I think it's fair to assume that the /2 was accidently ommited.
 
  • #7
Yes, but I'm just dang picky!
 

What is the equation of a sphere?

The equation of a sphere is (x-h)^2 + (y-k)^2 + (z-z)^2 = r^2, where (h,k,z) is the center of the sphere and r is the radius.

How do you find the equation of a sphere?

To find the equation of a sphere, you need to know the center point and the radius. Then, you can plug those values into the equation (x-h)^2 + (y-k)^2 + (z-z)^2 = r^2 to get the final equation.

What is the significance of the equation of a sphere?

The equation of a sphere is significant because it represents a three-dimensional shape that is perfectly symmetrical in all directions. It is used in many fields of science, including physics, astronomy, and geometry.

Can the equation of a sphere be simplified?

No, the equation of a sphere cannot be simplified. It is a fundamental equation that accurately describes the shape of a sphere in three-dimensional space.

Are there any other ways to represent a sphere mathematically?

Yes, there are other ways to represent a sphere mathematically, such as using vector equations or parametric equations. However, the standard equation (x-h)^2 + (y-k)^2 + (z-z)^2 = r^2 is the most commonly used and recognized form.

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