Parametric coordinates of an ellipse

[chaoseverlasting]

Homework Statement

[tex] \frac{x^2}{a^2}+\frac{y^2}{a^2(1-e^2)} =1[/tex]

The ellipse meets the major axis at a point whose abscissa is [tex]\lambda[/tex]. Find lim [tex]\theta ->0[/tex].

Homework Equations

Parametric coordinates of an ellipse: (acosx,bsinx)

The Attempt at a Solution

The abscissa is the x coordinate and here the x-axis is the major axis as [tex] b^2 = a^2(1-e^2)[/tex]. Therefore, [tex]\lambda =acos\theta[/tex].

Which would give you [tex]a[/tex]. The answer, however, should be ae... don't know how...

 

[AKG]

Could you write the problem out exactly as it's given. I find it hard to believe they'd ask you to find [itex]\lim _{\theta \to 0}[/itex] without saying what you're supposed to be finding the limit of as theta goes to 0. Is it the limit of the abscissa of the point on the ellipse at angle theta to the real axis, as theta goes to 0?

 

[HallsofIvy]

I'm with AKG! Find the limit of WHAT as [itex]\theta[/itex] goes to 0?

 

[chaoseverlasting]

But that's the whole question. Limit of [tex]\lambda[/tex] as [tex]\theta[/tex] goes to 0. But [tex]\lambda[/tex] comes out to be a which is totally independent of theta.

 

[AKG]

No, write it out EXACTLY as given.

 

[HallsofIvy]

But that's the whole question. Limit of [tex]\lambda[/tex] as [tex]\theta[/tex] goes to 0. But [tex]\lambda[/tex] comes out to be a which is totally independent of theta.

That was NOT what you originally said. You said "Find lim [itex]\theta-> 0[/itex], NOT [itex]lim \lambda as \theta-> 0[/itex]!.

Now, tell us how [itex]\theta[/itex] is defined!

 

[chaoseverlasting]

What I've written out, IS the whole question. Thats all there was to it. This question came in an exam, and the answer I got is independent of theta. I don't see how it can be anything else! All they're asking for is the point of intersection of the ellipse and the x axis. And that's (a,0).

 

[AKG]

You sure you wrote out exactly what was given? They honestly asked you to find "[itex]\lim _{\theta \to 0}[/itex]"? They didn't tell you what to find the limit of? They didn't define [itex]\theta[/itex]? Assuming that 0 < 1 - e² < 1 (note that this an assumption, one that should have been made explicit in the question), the major axis is the line segment from (-a,0) to (a,0). It doesn't make sense to speak of the point where the ellipse meets the major axis because it meets it at two different places, (-a,0) and (a,0), and both these points have different abscissae. Thus [itex]\lambda[/itex] isn't even well-defined.

A question has to make sense, i.e. it has to mean something, before you can conceivably find a right answer to it. For the reasons above, this question is far from making sense. So if you're SURE that you've written out the problem EXACTLY as it's given to you, then there's nothing anyone can do to help you because the problem doesn't make sense. Otherwise, WRITE OUT THE PROBLEM EXACTLY AS GIVEN.

 

[chaoseverlasting]

Guess the problem doesn't make sense. Must have been a typo in the paper.