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Exploring Ellipses
and Hyperbolas (JavaSketchpad)
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:
- Create a circle somewhere on the page.
- Construct a point on this circle. Make sure that it moves freely and does
not resize the circle.
- 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.
- 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.
- 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.
- 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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Tips and Tricks for Working with
JavaSketchpad
- 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.
If you have an other comments please contact
me
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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
- A locus of points is a collection of points or other objects
that satisfies a particular requirement.
- A 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
- Download GSP File
Hyperbola
A Hyperbola is constructed the exact same way as an ellipse except the second
focus goes OUTSIDE the circle not inside.