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Hyperbola equation


Members: Sunny Marcas Walter Sam Roxanne

Reviewing
From the last period , we have known that :
A hyperbolas is the set of all the points in a plane whose distance from t

wo fixed points in the plane have a constant difference.

Focus Vertex

Center

Focus Vertex

Figure shows a hyperbola centered at the origin with its focal axis on the x-axis. The vertices are at (-a, 0) and (a, 0), where a is some positive constant. The fixed points F1 (-c, 0) and F2 ( c, 0) are the foci of the hyperbola, with c>a

Notice that the hyperbola has two branches. For a point P( x, y) on the right-hand branch, PF1 - PF2 = 2a

On the left-hand branch, PF2 - PF1 = 2a . Combining these two equations gives us

PF1 - PF2 = ±2a

? PF 1 ? PF 2 ? ?2a
Using the distance formula, the equation becomes.

(x ? c)2 ? (y - 0) 2 ? ( x ? c) 2 ? ( y ? 0) 2 ? ?2a ( x ? c ) 2 ? y 2 ? ?2 a ? ( x ? c ) 2 ? y 2 x 2 ? 2cx ? c 2 ? y 2 ? 4a 2 ? 4a ( x ? c) 2 ? y 2 ? x 2 ? 2cx ? c 2 ? y 2 ? a ( x ? c) 2 ? y 2 ? a 2 ? c 2 x 2 a 2 ( x 2 ? 2cx ? c 2 ? y 2 ) ? a 4 ? 2a 2 cx ? c 2 x 2 (c 2 ? a 2 ) x 2 ? a 2 y 2 ? a 2 (c 2 ? a 2 )

The number a is the semitransverse axis, and b is the semiconjugate axis.
The hyperbola x2 y2 ? 2 ?1 2 a b
x2 y2 x2 y2 b ? 2 ?1? 2 ? 2 ? 0 ? y ? ? x 2 a b a b a
hyperbola

Replace 1 by 0

asymptotes

has two asympotptes. These asympotoes are slant lines that can be found by replacing the 1 on the right-hand side of the hyperbola's equation by 0:

Letting

b2 ? c2 ? a 2 b 2 x 2 ? a 2 y 2 ? a 2b 2
Its focal axis of hyperbola is on x-axis.

which is usually wirtten as

x2 y2 ? 2 ?1 2 a b y2 x2 ? 2 ?1 2 a b

Its focal axis of hyperbola is on y-axis.


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