# Defining an ellipse with the major axis and a point

1. Jun 28, 2012

### Luan Potter

Hello! I'm not sure this is the right place to ask, but i have a huge problem. I have an axis and it's origin, and also a point. I suposed that is enought to describe an ellipse, but how to do so? I want to describe it by having it's center and both it's radius. Some images to ilustrate what I am trying to say:
http://img855.imageshack.us/img855/4003/img1b.png [Broken]
Here you can see what I have: the red point and the black rect, and also the orange point (I don't know where in the black rect the ellipse is going to end, so I don't know it's diameter).
I think it's pretty simple the way I want to describe it, but it's very uncommom, so:
http://img266.imageshack.us/img266/3527/img2k.png [Broken]
I want the red center point (x, y) and the numeric values of each radius (the blue and the yellow ones).
I'd be really apreciated if anyone could help me with this problem. I've been looking for it for a long time now, but I just coudn't find an answer.

Last edited by a moderator: May 6, 2017
2. Jun 28, 2012

### HallsofIvy

Frankly, your pictures don't make a whole lot of sense to me.

But if your ellipse has center at $(x_0, y_0)$ in some coordinate system, and one axis of symmetry is parallel to the x-axis and has length 2a while the other is parallel to the y-axis and has length 2b, then the equation of the ellipse is
$$\frac{(x- x_0)^2}{a^2}+ \frac{(y- y_0)^2}{b^2}= 1$$

If you know the length of one axis, say the 2a, and know one point on the ellipse, say $(x_1, y_1)$, you can put that into the equation to get
$$\frac{(x_1-x_0)^2}{a^2}+ \frac{(y_1-y_0)^2}{b^2}= 1$$
and solve that for b.

3. Jun 28, 2012

### Luan Potter

Thank you for your answer, but there something missing still. I don't know the length of the axis (it is, though, parallel to the x-axis). I know it's starting point and the direction, but I don't know the end and neither the length (and, therefore, neither the center of the ellipse). From the first picture, imagine I have the red dot and the axis (but I don't know where it's going to end). With the orange dot, however, I can assume where the ellipse is going to be (I drew half of the ellipse with blue where I guessed it would have to be). But that is supposition, I needed to know the right formula. Maybe it's not possible, I don't know, but I can see only one possible ellipse that has this two pre-requisits). Sorry if it was hard to understand, I hope this has clarified it.

EDIT: I found out the mistake; with the informations given, there were infinity possibilities of ellipses. I looked my problem more deeply and found another information, and then successfully use your formula to solve the problem. It worked greatly! Thanks again for your help, and sorry for my mistake.

Last edited: Jun 28, 2012