# How do I find the width of a given section of an ellipse?

How would I be able to find the width of a horizontal ellipse (with a major axis of 120 and a minor axis of 5) at any given point along the major axis?

• Can you please define what you mean by width? Commented Mar 2, 2014 at 3:32
• By width I mean the length of a segment between the top and bottom curves of the ellipse that intersects and is perpendicular to the major axis. For example, if the ellipse were a wing then the width would be the chord length. Commented Mar 2, 2014 at 4:02

## 1 Answer

To solve your question you just need to write the equation of the ellipse. The most natural coordinates for writing the equation are the ones where the origin coincides with the center of the ellipse and the major and minor axes are along the $x$ and $y$-axes respectively. In this coordinate system, the equation of your ellipse is $x^2/(60)^2+y^2/(2.5)^2=1$. Now given a point $(a,0)$ on the major axis, all you need to do is find the values of $y$ which satisfy the equation $a^2/(60)^2+y^2/(2.5)^2=1$. The difference of these values of $y$ is the width of the ellipse at $(a,0)$. Can you see why?