Why is b^2 = 4 for this ellipse?

[Darkmisc]

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

 

[fresh_42]

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!

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?

View attachment 302317

 

[Steve4Physics]

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.

 

[Darkmisc]

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.

 

[fresh_42]

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.

 

[epenguin]

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 x_max are defined, but you need another condition else the min height and area are 0.

 

[Steve4Physics]

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.