Given a general quadratic curve
(1)
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the quantity is known as the discriminant, where
(2)
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and is invariant under rotation. Using the coefficients from quadratic equations for a rotation by an angle ,
(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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Now let
(9)
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(10)
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(11)
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(12)
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and use
(13)
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(14)
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to rewrite the primed variables
(15)
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(16)
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(17)
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(18)
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From (16) and (18), it follows that
(19)
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Combining with (17) yields, for an arbitrary
(20)
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(21)
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(22)
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(23)
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which is therefore invariant under rotation. This invariant therefore provides a useful shortcut to determining the shape represented by a quadratic curve. Choosing to make (see quadratic equation), the curve takes on the form
(24)
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Completing the square and defining new variables gives
(25)
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Without loss of generality, take the sign of to be positive. The discriminant is
(26)
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Now, if , then and both have the same sign, and the equation has the general form of an ellipse (if and are positive). If , then and have opposite signs, and the equation has the general form of a hyperbola. If , then either or is zero, and the equation has the general form of a parabola (if the nonzero or is positive). Since the discriminant is invariant, these conclusions will also hold for an arbitrary choice of , so they also hold when is replaced by the original . The general result is
1. If , the equation represents an ellipse, a circle (degenerate ellipse), a point (degenerate circle), or has no graph.
2. If , the equation represents a hyperbola or pair of intersecting lines (degenerate hyperbola).
3. If , the equation represents a parabola, a line (degenerate parabola), a pair of parallel lines (degenerate parabola), or has no graph.