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Unformatted text preview: the conic sections is the locus. However, if it is, it must arise from rotation of one of the model versions above. We will see that the locus of every instance of Equation ( 3.35 ) with not all leading terms zero has a locus ±tting one of these descriptions (with diFerent centers, foci and directrices), or a degenerate locus (line, point or empty set). To this end, we shall start from Equation ( 3.35 ) and show that in an appropriate coordinate system the equation ±ts one of the molds above. Let us denote the polynomial on the left side of Equation ( 3.35 ) by p ( x,y ): p ( x,y ) = Ax 2 + Bxy + Cy 2 + Dx + Ey. Assuming they don’t all vanish, the leading terms de±ne a quadratic form Q ( x,y ) = Ax 2 + Bxy + Cy 2 with matrix representative A = [ Q ] = b A B/ 2 B/ 2 C B ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.
- Spring '08