One property common to each of these curves is that it will intersect any given line at most 2 times. Non-degenerate quadratic plane curves are often called conic sections, since each can be obtained by slicing a double-napped cone.
The equation of each conic section is a second-degree polynomial function of two
variables, and can be written in the form
One property common to each of these curves is that it will intersect any given
line at most 2 times. Non-degenerate quadratic plane curves are often
called conic sections, since each can be obtained by slicing a
double-napped cone.
Each of the graphs on this page is a quadratic plane curve, displayed in the
window
.
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 An ellipse is one type of quadratic plane curve, or conic section. |
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 Ellipses can be oriented in any direction. They do not have to be symmetric with either axis. |
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 A circle is also a quadratic plane curve, or conic section. A circle can also be viewed as a very special ellipse. |
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 When the radius of a circle shrinks to zero, a degenerate circle will result. In other words, a single point is one type of quadratic plane curve. |
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 Quadratic equations in two variables may have no real solutions. Examining the equation, you might be inclined to describe this as a circle with an imaginary radius. |
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 A hyperbola is also a quadratic plane curve, or conic section. |
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 Hyperbolas can be functions, if one of the asymptotes is vertical. |
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 Here is a degenerate hyperbola, which will result when the constant term of a hyperbola is changed to zero. |
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 A parabola is also a quadratic plane curve, or conic section. |
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 Quadratic equations in two variables can also produce the degenerate case of two parallel lines. |
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 Another degenerate case of a quadratic equation will produce a single line, though with multiplicity two. This occurs when the equation is a perfect square on one side, and zero on the other. |