# How to draw an ellipse with its center and two points on it with with Sketchpad?

How to draw an ellipse with its center and two points on it with Sketchpad?

## 3 Answers

More than one ellipse can share the same center and pass through two random points A and B.

To find out how to construct ellipses with Sketchpad, check out my book, Exploring Conic Sections with The Geometer's Sketchpad.

Charles, perhaps you meant to consider only ellipses with their axes oriented horizontally and vertically. If so, the simplest approach is probably to translate the coordinate system so the ellipse center is at the origin, and then to substitute each through-point's coordinates into x^2/a^2 + y^2/b^2 = 1. The result is two simultaneous equations in a and b, the solution to which determines the lengths a and b of the semi-axes.

Once you know a and b, the easiest way to graph the ellipse is to use the parametric functions x(theta) = a cos(theta) and y(theta) = b sin(theta). Specifically, create (but do not graph) these two functions, select them, and choose Graph | Plot Parametric Curve. You'll need to set the domain to go from 0 to 2 pi.

The two through-points cannot be aligned horizontally or vertically. In addition, they must be oriented between 90° and 180° relative to each other, or between 270° and 360°; otherwise they determine a hyperbola instead of an ellipse.

Charles - Scott has provided a thorough analytic answer to your question. I took a more geometric approach, trying to figure out how to construct the second focus from the three given points, so that I could use the Sketchpad's custom ellipse tool (which needs both foci as a given). I've posted my sample sketch, diagram, and description of what I did here at Key Curriculum's free Sketch Exchange site.

draks and J.M. - Sketchpad is a dynamic algebra and geometry software – click here for more information.

• This is close, but in the question it is the center of the ellipse, midway between both foci, that is given. – Rahul Jan 5 '13 at 22:40