Hyperbola
Definition. The locus of the points for which the difference of distances to two fixed points is constant equal to 2а is called a hyperbola. As for an ellipse, we introduce a new coordinate system:
М (х,у)
F 1(–c,0) M 1 M 2 F 2(c,0) x
To derive the equation of a hyperbola, we take an arbitrary point М(х,у) on the hyperbola and consider the distances from this point to the foci:
The characteristic feature of the line is, by definition,
We have composed an equation of the hyperbola. Let us reduce to a convenient form (by analogy with the ellipse):
We divide both sides by
Changing the sign, we obtain the equation of a hyperbola:
Since 2 a< 2 c, we denote the difference of squares by
Thus, we have obtained the classical equation of a hyperbola:
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у
r 1 r 2
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