# 12.2 The hyperbola  (Page 2/13)

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## Deriving the equation of an ellipse centered at the origin

Let $\text{\hspace{0.17em}}\left(-c,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(c,0\right)\text{\hspace{0.17em}}$ be the foci    of a hyperbola centered at the origin. The hyperbola is the set of all points $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ such that the difference of the distances from $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ to the foci is constant. See [link] .

If $\text{\hspace{0.17em}}\left(a,0\right)\text{\hspace{0.17em}}$ is a vertex of the hyperbola, the distance from $\text{\hspace{0.17em}}\left(-c,0\right)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}\left(a,0\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}a-\left(-c\right)=a+c.\text{\hspace{0.17em}}$ The distance from $\text{\hspace{0.17em}}\left(c,0\right)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}\left(a,0\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}c-a.\text{\hspace{0.17em}}$ The sum of the distances from the foci to the vertex is

$\left(a+c\right)-\left(c-a\right)=2a$

If $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ is a point on the hyperbola, we can define the following variables:

By definition of a hyperbola, $\text{\hspace{0.17em}}{d}_{2}-{d}_{1}\text{\hspace{0.17em}}$ is constant for any point $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ on the hyperbola. We know that the difference of these distances is $\text{\hspace{0.17em}}2a\text{\hspace{0.17em}}$ for the vertex $\text{\hspace{0.17em}}\left(a,0\right).\text{\hspace{0.17em}}$ It follows that $\text{\hspace{0.17em}}{d}_{2}-{d}_{1}=2a\text{\hspace{0.17em}}$ for any point on the hyperbola. As with the derivation of the equation of an ellipse, we will begin by applying the distance formula    . The rest of the derivation is algebraic. Compare this derivation with the one from the previous section for ellipses.

This equation defines a hyperbola centered at the origin with vertices $\text{\hspace{0.17em}}\left(±a,0\right)\text{\hspace{0.17em}}$ and co-vertices $\text{\hspace{0.17em}}\left(0±b\right).$

## Standard forms of the equation of a hyperbola with center (0,0)

The standard form of the equation of a hyperbola with center $\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and transverse axis on the x -axis is

$\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$

where

• the length of the transverse axis is $\text{\hspace{0.17em}}2a$
• the coordinates of the vertices are $\text{\hspace{0.17em}}\left(±a,0\right)$
• the length of the conjugate axis is $\text{\hspace{0.17em}}2b$
• the coordinates of the co-vertices are $\text{\hspace{0.17em}}\left(0,±b\right)$
• the distance between the foci is $\text{\hspace{0.17em}}2c,$ where $\text{\hspace{0.17em}}{c}^{2}={a}^{2}+{b}^{2}$
• the coordinates of the foci are $\text{\hspace{0.17em}}\left(±c,0\right)$
• the equations of the asymptotes are $\text{\hspace{0.17em}}y=±\frac{b}{a}x$

See [link] a .

The standard form of the equation of a hyperbola with center $\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and transverse axis on the y -axis is

$\frac{{y}^{2}}{{a}^{2}}-\frac{{x}^{2}}{{b}^{2}}=1$

where

• the length of the transverse axis is $\text{\hspace{0.17em}}2a$
• the coordinates of the vertices are $\text{\hspace{0.17em}}\left(0,±a\right)$
• the length of the conjugate axis is $\text{\hspace{0.17em}}2b$
• the coordinates of the co-vertices are $\text{\hspace{0.17em}}\left(±b,0\right)$
• the distance between the foci is $\text{\hspace{0.17em}}2c,$ where $\text{\hspace{0.17em}}{c}^{2}={a}^{2}+{b}^{2}$
• the coordinates of the foci are $\text{\hspace{0.17em}}\left(0,±c\right)$
• the equations of the asymptotes are $\text{\hspace{0.17em}}y=±\frac{a}{b}x$

See [link] b .

Note that the vertices, co-vertices, and foci are related by the equation $\text{\hspace{0.17em}}{c}^{2}={a}^{2}+{b}^{2}.\text{\hspace{0.17em}}$ When we are given the equation of a hyperbola, we can use this relationship to identify its vertices and foci.

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