A hyperbola consists of two smooth branches. The turning point of each branch is a vertex of the hyperbola. The segment connecting the two vertices is the transverse axis, which lies on the axis of symmetry. The two foci also lie on the axis of symmetry. The center of the hyperbola is the midpoint between the two vertices, which also is the midpoint between the two foci.

Just as for an ellipse, if the foci are ( ± c , 0 ) , open plus minus c comma 0 close comma  the distance between the two foci is 2c. If the vertices are ( ± a , 0 ) , open plus minus eh comma 0 close comma  the distance between the vertices is 2a.

Since vertex P is on the hyperbola, it must satisfy the equation | P F 1 − P F 2 | = k , vertical line p , f sub 1 , minus p , f sub 2 , vertical line equals k comma  but you can also see that

Image Long Description

| P F 1 − P F 2 | = | [ 2 a + ( c − a ) ] − ( c − a ) |   = | 2 a + c − a − c + a |   = | 2 a | = 2 a table with 3 rows and 2 columns , row1 column 1 , absolute value of p , f sub 1 , minus p , f sub 2 , end absolute value , , column 2 equals vertical line left bracket 2 eh plus open c minus eh close right bracket minus open c minus eh close vertical line , row2 column 1 , , column 2 equals vertical line 2 eh plus c minus eh minus c plus eh vertical line , row3 column 1 , , column 2 equals vertical line 2 eh vertical line equals 2 eh , end table

Therefore, k = 2 a . k equals 2 eh .

In a standard hyperbola, c is related to a and b by the equation c 2 = a 2 + b 2 . c squared , equals , eh squared , plus , b squared , .  The length of the conjugate axis is 2b. The transverse and conjugate axes determine a rectangle that lies between the vertices, and the diagonals of that central rectangle determine the asymptotes of the hyperbola. Recall that an asymptote is a line that a graph approaches. The branches of the hyperbola will approach the asymptotes.

Take Note Key Concept: Properties of Hyperbolas with Center (0, 0)

Image Long Description Horizontal Hyperbola	Image Long Description Vertical Hyperbola
Equation: x 2 a 2 − y 2 b 2 = 1 fraction x squared , over eh squared end fraction . minus . fraction y squared , over b squared end fraction . equals 1	Equation: y 2 a 2 − x 2 b 2 = 1 fraction y squared , over eh squared end fraction . minus . fraction x squared , over b squared end fraction . equals 1
Transverse axis: Horizontal	Transverse axis: Vertical
Vertices: ( ± a , 0 ) open plus minus eh comma 0 close	Vertices: ( 0 , ± a ) open 0 comma plus minus eh close
Foci: ( ± c , 0 ) , open plus minus c comma 0 close comma  where c 2 = a 2 + b 2 c squared , equals , eh squared , plus , b squared	Foci: ( 0 , ± c ) , open 0 comma plus minus c close comma  where c 2 = a 2 + b 2 c squared , equals , eh squared , plus , b squared
Asymptotes: y = ± b a x y equals plus minus , b over eh , x	Asymptotes: y = ± a b x y equals plus minus , eh over b , x

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