Fun with Conic Sections
We will begin with the general equation of a conic section: [math](x-h)^2/a + (y-k)^2/b = r^2[/math][br][br]There are five parameters here: [math]a, b, h, k, r[/math][br][br]We will see how the equation and shape of a conic section changes with changes in each of these parameters.
[b][color=#c51414]Move the sliders around to observe changes in the shape of the graph. Also note that each parameter is color coded in the equation on the right. Changing the parameter hence also shows you changes in the equation.[/color] [br][/b][br]You can now answer some interesting questions![br][list=1][br][*]Which parameters determine the '[i]shape[/i]' of your graph? When is your graph a circle? An ellipse? A hyperbola?[br][*]When is your ellipse vertical? When is it horizontal?[br][/list]
General Conic Section - Which cone/plane create the curve?
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For any general conic section defined by Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, this applet gives you a cone and a plane whose intersection gives the same curve. For non-degenerate cases, the type of conic curve is identified (ellipse, parabola, hyperbola). Admittedly, degenerate cases (points, parallel lines, intersecting lines) are not detected, yet the world keeps turning! |
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