12.2 The hyperbola  (Page 4/13)

Standard forms of the equation of a hyperbola with center ( h , k )

The standard form of the equation of a hyperbola with center $\left(h,k\right)$ and transverse axis parallel to the x -axis is

where

• the length of the transverse axis is $2a$
• the coordinates of the vertices are $\left(h\pm a,k\right)$
• the length of the conjugate axis is $2b$
• the coordinates of the co-vertices are $\left(h,k\pm b\right)$
• the distance between the foci is $2c,$ where $c^2=a^2+b^2$
• the coordinates of the foci are $\left(h\pm c,k\right)$

The asymptotes of the hyperbola coincide with the diagonals of the central rectangle. The length of the rectangle is $2a$ and its width is $2b.$ The slopes of the diagonals are $\pm \frac{b}{a},$ and each diagonal passes through the center $\left(h,k\right).$ Using the point-slope formula , it is simple to show that the equations of the asymptotes are $y=\pm \frac{b}{a}\left(x-h\right)+k.$ See [link] a

The standard form of the equation of a hyperbola with center $\left(h,k\right)$ and transverse axis parallel to the y -axis is

where

• the length of the transverse axis is $2a$
• the coordinates of the vertices are $\left(h,k\pm a\right)$
• the length of the conjugate axis is $2b$
• the coordinates of the co-vertices are $\left(h\pm b,k\right)$
• the distance between the foci is $2c,$ where $c^2=a^2+b^2$
• the coordinates of the foci are $\left(h,k\pm c\right)$

Using the reasoning above, the equations of the asymptotes are $y=\pm \frac{a}{b}\left(x-h\right)+k.$ See [link] b .