How do you find the focus of an ellipse, say in the form of its distance from the center of the ellipse. I had thought it could be

[(m-x)² +r² -(nx)² ]^1/2 +[(m+x)² +r² -(nx)² ]^1/2 =C

where

m is the distance of the focus from the origin (it lies on the x axis)

and

y² +n² x² =r²

for n less than one

and C is a constant.

this appears to yield m being dependent upon x. Does anyone know of a way to find the focus point of an ellipse with the above equation?

thanks,

Brad

One definition of an ellipse is the closed curve such that the sum of distances from the foci to the point on ellipse is constant. Applying this, we have:

2(c² +b² )^½ =(a+c)+(a-c)

where a and b are the semimajor and semiminor radii respectively, and c is the required distance(notice that the foci must lie on the semimajor axis). This yields the simple formula:

c² =a² -b²

for the ellipse (x/a)² +(y/b)² =1.

Kerwin

To ask a related question, is there a set formula for the circumference (if that is the right word in this case) for an ellipse given the cartesian equation of that ellipse?

I'd suspect that there is no simple formula (although there certainly is a formula) to find the circumference of an ellipse. If you work through the math, it leads to a horrendous integration, which I'm not sure how to do. Even if I did know a way to do the integration, the formula would be pretty complex.

Brad

The circumference of a circle of radius 1 is 2p.

If you have a cartesian ellipse ((x-r)/a)²+((y-s)/b)²=1 it has circumference 2a bp. The equation of any ellipse can be got into this form by rotation of the axes.

i.e. set

X=xcosq+ysinq

Y=ycosq = xsinq

and choose an appropriate q.

No, the area of an ellipse is pa b, but the circumference leads to an integral that can't be evaluated in terms of elementary functions (this can be proved, although I don't know how). It is called an elliptic integral, and is related to the area of mathematics Wiles used to prove Fermat's Last Theorem.

You may be interested in Ramanujan's approximation to the circumference of an ellipse.

The easiest way to prove the area of an ellipse is using the fact that an ellipse of semiaxes a and b is the image of a circle of radius 1 under the transformation defined by the matrix:

a 0
0 b

Which has determinant ab. The determinant of a transformation is the area scale factor, so the area of the ellipse is ab times the area of a circle of radius 1.

of course you're right I am a nitwit