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Hyperbola

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Let L{\displaystyle L} be the point on the line PF2¯{\displaystyle {\overline {PF_{2}}}} with the distance 2a{\displaystyle 2a} to the focus F2{\displaystyle F_{2}} (see diagram, a{\displaystyle a} is the semi major axis of the hyperbola). Line w{\displaystyle w} is the bisector of the angle between the lines PF1¯,PF2¯{\displaystyle {\overline {PF_{1}}},{\overline {PF_{2}}}}. In order to prove that w{\displaystyle w} is the tangent line at point P{\displaystyle P}, one checks that any point Q{\displaystyle Q} on line w{\displaystyle w} which is different from P{\displaystyle P} cannot be on the hyperbola. Hence w{\displaystyle w} has only point P{\displaystyle P} in common with the hyperbola and is, therefore, the tangent at point P{\displaystyle P}. From the diagram and the triangle inequality one recognizes that |QF2|<|LF2|+|QL|=2a+|QF1|{\displaystyle |QF_{2}|<|LF_{2}|+|QL|=2a+|QF_{1}|} holds, which means: |QF2|−|QF1|<2a{\displaystyle |QF_{2}|-|QF_{1}|<2a}. But if Q{\displaystyle Q} is a point of the hyperbola, the difference should be 2a{\displaystyle 2a}.

Since the eccentricity of a hyperbola is always greater than one, the center B must lie outside of the reciprocating circle C .(Hilbert and Cohn-Vossen 1999, p. 3). Letting fall on the left -intercept requires thatThis ratio is called the eccentricity , and for a hyperbola it is always greater than 1.


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