# Ellipse through three points

1. Jul 5, 2011

### Nelourir

1. The problem statement, all variables and given/known data

First of all, I'm not sure the following is achievable at my level (last year of high school), as it is not posed as a direct question, but if yes, could you point me in the right direction?

I'm trying to find an ellipse that goes through three points, (0,0) , (b,0) and (b/2,h).
This actually represents a building with a base of width b and a height of h.

The building is symmetrical, therefore the ellipse is not 'rotated' (I'm not sure how to express this mathematically).

3. The attempt at a solution
Using the equation for an ellipse (not trying to confuse anyone but h is not H and b is not B):
(((X-H)^2)/(A^2)) + (((Y-K)^2)/(B^2)),
I find the following (using logic and no algebra):

H is b/2 as the centre of the ellipse always goes through the line x=b/2
A is greater than or equal to b/2. (otherwise the ellipse cannot touch both points on the x axis)
B is greater than or equal to h. (I don't want the building to have angles greater than 90o)
Using the above statement, I expressed B as h + C and found that K = -C.

I realise there is an infinite number of ellipses (at least I'm assuming so), but I think that A should be dependent on b, h and/or C. Could you help me find this?

EDIT: OK, I think I found my answer by replacing A by b/2 + D, plugged in x=0, y=0 then I rearranged until it looked like a quadratic equation with D as the 'variable' and I got an answer for D in terms of b, h and D. Sorry for making a new topic :shy:.

Last edited: Jul 5, 2011