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The analytic equation for a conic in arbitrary position is the following:
where at least one of A, B, C is nonzero. To reduce this to one of the forms given previously, perform the following steps (note that the decisions are based on the most recent values of the coefficients, taken after all the transformations so far):
Now C=0. (This step corresponds to rotating and scaling about the origin.)
We work out an example for clarity. Suppose the original equation is
In step 1 we apply the substitutions x2x+y and y2y-x. This gives 25x+10x-5y+1=0. Next we interchange x and y (step 2) and get 25y+10y-5x+1=0. Replacing y by y- in step 3 we get 25y-5x=0. Finally, in step 4a we divide the equation by 25, thus giving it the form (3) with a=. We have reduced the conic to a parabola with vertex at the origin and focus at (,0). To locate the features of the original curve, we work our way back along the chain of substitutions (recall the convention about substitutions and transformations from Section 1.1):
We conclude that the original curve (3) is a parabola with vertex and focus .
If one just wants to know the type of the conic defined by (1) , an alternative analysis consists in forming the quantities
and finding the appropriate case in the following table, where an entry in parentheses indicates that the equation has no solution over the real numbers:
For the central conics (the ellipse, the point, and the hyperbola), the center (x,y) is the solution of the system of equations
2Ax+Cy+D=0,
Cx+2By+E=0,
and the axes have slope q and -1/q, where q is given by (2) .
Silvio Levy
Wed Oct 4 16:41:25 PDT 1995
This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press). Unauthorized duplication is forbidden.