The picture shows the standard hyperbola (gray) and its image (blue) through the transformation F: [x'=ax, y' =cx+by]. P' is the image of P. The asymptotic triangle OAB of the standard hyperbola is mapped to the asymptotic triangle OA'B' of the image-hyperbola. The area of OAB is 2 and that of OA'B' is 2 multiplied by the determinant of the matrix defining the transformation. This being ab, the area of the asymptotic triangle OA'B' is 2ab. The points of the hyperbola can be represented as the graph of the function y' = (c/a)x' + (ab)/x'.

HyperbolaRectangular.html

OrthoRectangular.html

RectangularAsProp.html

AsymptoticTriangle.html

HyperbolaAsymptotics.html

RectHyperbola.html

RectHypeRelation.html

Hyperbola.html

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