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As a result, we obtain
By definition,
y
M1(–à;0) 0 M2(à;0) x
Since the eccentricity of a hyperbola is The ratios of the focal radii to the corresponding distances from an arbitrary point of the hyperbola to the directrices is constant and equals the eccentricity:
Tangent lines to a hyperbola. Suppose given, the equation of a hyperbola and a point Ì0(õ0,ó0) on it:
As is known, the equation of a tangent to a curve is
Let us differentiate the equation of a hyperbola as an implicit function:
Substituting, we obtain the equation of the tangent line:
Let us transform it:
Since the point Ì0 belongs to the hyperbola, its coordinates must satisfy the equation of the hyperbola, and, the right-hand side equals (–1) . Thus, the equation of a tangent to a hyperbola is
Example. Write the equation of the hyperbola with real semi axis 4 and foci at the points F1(–5;0) and F2(5;0). Formula (19) gives We have The required classical equation of the hyperbola is
Date: 2015-01-02; view: 1182
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