Why is b^2 = 4 for this ellipse?

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In summary, the question is asking for the Cartesian equation of the smallest ellipse that passes through the point (6,0) and has a centre at (2,0). However, there seems to be a problem with the question as it is possible to make ellipses with smaller values of b that still pass through (6,0). Additional information or clarification may be needed to determine the correct solution.
  • #1
Darkmisc
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Homework Statement
Find the Cartesian equation of the smallest ellipse that passes through the point (6,0) and has a centre at (2,0).
Relevant Equations
x^2/a^2 + y^2/b^2 = 1
Hi everyone

The solution for this question has b^2 as 4, but I don't see why it has to be 4. I've tried using different values of b for the ellipse on Desmos, and it is possible to make ellipses with smaller values of b that pass through (6,0).

Have I missed something in the question? Or has the question omitted an assumption (e.g. that the ellipse has to have the same height as the circle)?Thanks

1654248266392.png
 
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  • #2
Darkmisc said:
Homework Statement:: Find the Cartesian equation of the smallest ellipse that passes through the point (6,0) and has a centre at (2,0).
Relevant Equations:: x^2/a^2 + y^2/b^2 = 1

Hi everyone

The solution for this question has b^2 as 4, but I don't see why it has to be 4. I've tried using different values of b for the ellipse on Desmos, and it is possible to make ellipses with smaller values of b that pass through (6,0).
Define 'smaller' in this context!
Darkmisc said:
Have I missed something in the question? Or has the question omitted an assumption (e.g. that the ellipse has to have the same height as the circle)?Thanks
Maybe you should describe in words what the smallest ellipse will be! Where on the ellipse must ##(6,0)## lie?
Darkmisc said:
View attachment 302317
 
  • #3
Darkmisc said:
The solution for this question has b^2 as 4, but I don't see why it has to be 4. I've tried using different values of b for the ellipse on Desmos, and it is possible to make ellipses with smaller values of b that pass through (6,0).

View attachment 302317
You are correct There seems to be a problem with the question (Post #1 attachment).

I guess the question should also require that the ellipse fully encloses the circle, or something equivalent to this.

(Also, when answering questions like this, I would include a diagram showing axes, circle and ellipse.)

Edit - typo's.
 
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  • #4
fresh_42 said:
Define 'smaller' in this context!

Maybe you should describe in words what the smallest ellipse will be! Where on the ellipse must ##(6,0)## lie?
The width of the ellipse would be constrained by having to pass through (6,0), so smaller to me would refer to its height. I'm not sure there's a limit to how flat the ellipse could be. I think b could be made arbitrarily small and the ellipse would still pass through (6,0), so long as a^2 = 16.
 
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  • #5
Darkmisc said:
The width of the ellipse would be constrained by having to pass through (6,0), so smaller to me would refer to its height. I'm not sure there's a limit to how flat the ellipse could be. I think b could be made arbitrarily small and the ellipse would still pass through (6,0), so long as a^2 = 16.
Yes, I imagined a flat line from ##(6,0)## to ##(-2,0),## too. So either @Steve4Physics is right and it was a typo, or some additional information is missing. I won't care very much.
 
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  • #6
Darkmisc said:
The width of the ellipse would be constrained by having to pass through (6,0), so smaller to me would refer to its height. I'm not sure there's a limit to how flat the ellipse could be. I think b could be made arbitrarily small and the ellipse would still pass through (6,0), so long as a^2 = 16.
I think you're right, the centre and xmax are defined, but you need another condition else the min height and area are 0.
 
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  • #7
fresh_42 said:
So either @Steve4Physics is right and it was a typo, or some additional information is missing.
For accuracy, can I note that (in Post #3) I didn't suggest there was a typo' in the question. I suggested additional information was missing.
 
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1. Why is b^2 = 4 for this ellipse?

The value of b^2 = 4 for this ellipse because it is a special case of an ellipse known as a circle. In a circle, the distance from the center to any point on the edge is the same, which means the length of the semi-minor axis (b) is equal to the length of the semi-major axis (a). Since the equation for a circle is x^2 + y^2 = r^2, where r is the radius, in this case, b^2 = a^2 = r^2. Since we know that the radius of a circle is equal to its diameter (which is twice the radius), we can substitute 2r for a and b, giving us b^2 = (2r)^2 = 4r^2 = 4.

2. How does b^2 = 4 affect the shape of the ellipse?

The value of b^2 = 4 affects the shape of the ellipse by making it a perfect circle. Since b^2 = 4, this means that the length of the semi-minor axis is equal to the length of the semi-major axis, resulting in a circle. If b^2 were to have any other value, the ellipse would be stretched or compressed, resulting in an oval shape.

3. Can b^2 ever be greater or less than 4 for an ellipse?

Yes, b^2 can be greater or less than 4 for an ellipse. In fact, for any ellipse that is not a circle, b^2 will have a different value than 4. This is because the semi-minor axis (b) is always shorter than the semi-major axis (a) in an ellipse, resulting in a value of b^2 that is less than a^2. However, in special cases where the ellipse is a perfect circle, b^2 will be equal to 4.

4. What happens if b^2 is negative for an ellipse?

If b^2 is negative for an ellipse, this means that the ellipse does not exist. This is because the value of b^2 represents the square of the length of the semi-minor axis, and a negative value cannot be squared to give a positive value. Therefore, an ellipse with a negative value for b^2 is not a valid geometric shape.

5. How does b^2 = 4 relate to the eccentricity of the ellipse?

The value of b^2 = 4 is directly related to the eccentricity of the ellipse. The eccentricity of an ellipse is a measure of how "stretched" or "compressed" the ellipse is, with a value of 0 representing a perfect circle and a value of 1 representing a line. Since b^2 = 4 results in a perfect circle, the eccentricity of this ellipse is 0. In general, the closer b^2 is to 4, the closer the ellipse is to a perfect circle, and the lower the eccentricity will be.

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