Calc III: minimize the area of an ellipse

[Vols]

1. Problem: ellipse x^2/a^2 + y^2/b^2 = 1 encloses circle x^2 + y^2 = 2x. Find values of a and b that minimize the area of the ellipse.

Homework Equations

: [/B]A = pi*a*b for an ellipse.

The Attempt at a Solution

: [/B]I tried a bunch of crazy stuff... I know I need to find where the tangents (1st derivatives) are equal for the ellipse and circle (at two intersections where x's are equal and y's will be mirrored across the x axis). Or I think I know that... I can do the implicit differentiation to get me the dy/dx for the ellipse and circle [-xb^2/ya^2 and (1-x)/y respectively] but I set them equal and don't really get anything out of it, so I tried solving the ellipse equation for y^2 [y^2=b^2/a^2(a^2-x^2)], plugging that into the circle equation, then getting dy/dx but that didn't seem to work either. I felt like I was making progress several times but now I just feel like I'm going in circles (or ellipses)... I can show much more of what I tried if it helps. I am still working on it as I post this.

P.S. I just looked over an old (2008) thread here about a very similar problem which seems to be helping but I still can't quite get there with what I just read.

 

[mfb]

You can calculate the point where circle and ellipse meet as function of a and b. dy/dx should be the same for ellipse and circle at that point, which gives you one condition on a and b.

 

[SteamKing]

1. Problem: ellipse x^2/a^2 + y^2/b^2 = 1 encloses circle x^2 + y^2 = 2x. Find values of a and b that minimize the area of the ellipse.

Homework Equations

: [/B]A = pi*a*b for an ellipse.

The Attempt at a Solution

: [/B]I tried a bunch of crazy stuff... I know I need to find where the tangents (1st derivatives) are equal for the ellipse and circle (at two intersections where x's are equal and y's will be mirrored across the x axis). Or I think I know that... I can do the implicit differentiation to get me the dy/dx for the ellipse and circle [-xb^2/ya^2 and (1-x)/y respectively] but I set them equal and don't really get anything out of it, so I tried solving the ellipse equation for y^2 [y^2=b^2/a^2(a^2-x^2)], plugging that into the circle equation, then getting dy/dx but that didn't seem to work either. I felt like I was making progress several times but now I just feel like I'm going in circles (or ellipses)... I can show much more of what I tried if it helps. I am still working on it as I post this.

P.S. I just looked over an old (2008) thread here about a very similar problem which seems to be helping but I still can't quite get there with what I just read.

Start with making a sketch of the circle and a tentative ellipse. You'll have to do a little algebra to figure out the center and radius of the circle.

 

[Vols]

I can't see where the radius/center of the circle play into finding a and b.

Also: I got y^2=b^2/a^2(a^2-x^2) and plugged that into the circle equation to get 2x - 2b^2x/a^2 - 2 = 0 in an attempt to get things in terms of a and b but it doesn't seem to be the right track.

 

[SteamKing]

I can't see where the radius/center of the circle play into finding a and b.

Also: I got y^2=b^2/a^2(a^2-x^2) and plugged that into the circle equation to get 2x - 2b^2x/a^2 - 2 = 0 in an attempt to get things in terms of a and b but it doesn't seem to be the right track.

Except the ellipse and the circle don't share a common center. This is something a sketch would make clear.

 

[Vols]

The first thing I did was pull it up on desmos. Still not sure how it helps to know that.

I get that it is significant to know that in order to solve. But I don't see how it actually is used to calculate a and b.

 

[Vols]

OK. I just found out that I missed the whole section on lagrange multipliers. Knowing that this was missing from my arsenal makes me feel a bit better, but, having only just read about them, I'm still having a bit of trouble. I want to minimize my objective function for area, f(a,b)=a*b*pi, but I'm not sure about my constraints or how to get them in terms of a and b. I know the ellipse and circle share 2 points with the same x values (b/w .5 and 1) and that the tangents at those points will have the same slope for both the ellipse and the circle. Also, the ellipse equation must be constrained to outside (or on) the circle at all points but that is a constraint on the wrong equation. I think with a bit of guidance I can understand the application of lagrange multipliers conceptually and figure out what to do, but the book is just horribly technical and isn't giving me the insight I'm after.

 

[mfb]

SteamKing: I don't understand what your advice is aiming for. Vols works with the equations, those equations are satisfied and you don't have to know centers of objects or anything else.

I know the ellipse and circle share 2 points with the same x values (b/w .5 and 1)

Where does that range come from?

I can do the implicit differentiation to get me the dy/dx for the ellipse and circle [-xb^2/ya^2 and (1-x)/y respectively] but I set them equal and don't really get anything out of it

You should get x of the intersection point as function of a and b.
By setting y² from the ellipse equal to y² for the circle you get an equation that contains x, a and b only. You can use the expression for x gained above and plug it in, it will give you an equation with a and b only. Solve for one of them, plug it into the formula for the area, and you have a one-dimensional minimization problem.

 

[Vols]

Holycrapthankyou! I have the equations done from previous attempts, I just need to sort through and plug everything in the right places. I will report back after I get a chance to do that - so much to do, so little time - but it makes sense to me sitting here right now (subject to change, ha) so I just wanted to say thanks, mfb.

EDIT: Where I said b/w .5 and 1 I meant b/w 1 and 2.