The activity that is presented here offers an opportunity to explore the properties of Ellipse and Hyperbolas through a geometric perspective within your browser window.

There are five areas to explore within this activity:

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Directions for Creating and Simulating an Ellipse in the JavaWindow:

1. Create a circle somewhere on the page.
2. Construct a point on this circle. Make sure that it moves freely and does not resize the circle.
3. Construct a point circle this will be the second of two foci. The first is the center of the circle. Now we need to construct a perpendicular bisector of a segment connecting our second foci to the point on the circle.
4. We can trace this line (click the appropriate button in the window). Now drag the point on the cirlce around and watch the formation that appears. This appears to be an ellipse. Clear the screen using the red X in the bottom right of the JavaWindow. Now move the second focus point further or closer from the center (the first focus) (click the red X again to clear the traced lines) and drag the point on the circle again or press the show animate button-button and try the animation.
5. We recall the definition of an Ellipse and we try to show that this is indeed a construction of an Ellipse. Click the button that creates the locus point to represent all the points of the locus. Drag or animate the sketch again and notice where that point is. Click the Show measurements and Construct trace button. Notice that the distance from the focus to the point of the locus is equal to the distance from the locus point to the point on the circle. (The measurements should be equal throughout the animation or movement). Check it for a few points. This is an important part of the definition.
6. Try to formulate a proof that this sketch is indeed an Ellipse.

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Constructing A Ellipse and Hyperbola (Java Window)

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• Remember that clicking the red X clears all the traces
• Try pressing return in the location bar at the top of your browser to reload the entire window (clear it all)
• Dragging the red points will result in changing the picture that the window shows you. Experiment with moving lines and points.
• Try to maximize the size of your browser window to see the entire picture
• The first time you load this page may take a few moments but wait and it will work fine.
• Clicking the Animate button once starts the animation. Clicking it again stops the animation.

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Questions and Explorations

• What does the perpendicular bisector represent when this activity is compared to the paper folding activity?
• What happens to the Ellipse when the focus is further away from the circle? What about closer to the circle? How about the center of the circle?
• What do the traced lines represent in relation to the Ellipse that is formed?
• How can you construct a proof that this sketch does indeed represent an Ellipse? (Hint look at the triangles formed from the vertices of the focus, locus point, and point on the circle)

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Definitions and Assistants

• locus of points is a collection of points or other objects that satisfies a particular requirement.
• focus is the point used to determine the Ellipse and Hyperbola's openness.
• An Ellipse is the locus of all points such that the sum of the distances from these points is always a constant.
• A Hyperbola is the locus of all points such that the difference of the distances from these points is always a constant