** CONIC SECTIONS: THE HYPERBOLA**

Example 1:

Find the coordinates of the center, vertices, and foci of the hyperbola given by . The equation is already in standard form

where

,h = 2,k = 5(from denominator of positive term!), anda = 4b = 5Now, we can find the requested information as follows:

Coordinates of the Center(h, k)

(2, 5)

Coordinates of the Vertices(h + a, k)and(h a, k)

and(2 + 4, 5)(24, 5)then

and(6, 5)(2, 5)

Coordinates of the Foci(h + c, k)and,(h c, k)whereSince

anda = 4, then .b = 5and

Example 2:

Find the coordinates of the center, vertices, and foci of the hyperbola given by .

First, we will convert the equation of the hyperbola to standard form. Since there is a positive

y-term raised to the second power we know that this hyperbola is of the form .^{2}NOTE: The center

of this hyperbola is at(h, k)because there is only a(0, 0)y-term and an^{2}x-term. Therefore, we can write^{2}where

,h = 0,k = 0(from denominator of positive term!), anda = 5b = 4Now, we can find the requested information as follows:

Coordinates of the Center(h, k)

(0, 0)

Coordinates of the Verticesand(h, k + a)(h, ka)

and(0, 0 + 5)(0, 05)then

and(0, 5)(0,5)

Coordinates of the Foci(h, k + c)and(h, k,c)whereSince

anda = 5, then .b = 4and

then and

Example 3:

Find the coordinates of the center, vertices, foci, and the equations of the asymptotes of the hyperbola given by .

First, we will convert the equation of the hyperbola to standard form.

It is either or , where

ora = ba bLet's divide both sides of the equation by

to get12

and

or

We can now write the standard form of the given hyperbolic equation as follows:

where

,h =1,k =0(from denominator of positive term!), anda = 2b =Now, we can find the requested information as follows:

Coordinates of the Center(h, k)

(1, 0)

Coordinates of the Verticesand(h, k + a)(h, ka)

and(1, 0 + 2)(1, 02)

and(1, 2)(1,2)

Coordinates of the Foci(h, k + c)and(h, k,c)whereSince

anda = 2, then .b =and

and

Example 4:

Find the coordinates of the center, vertices, and foci of the hyperbola given by .

First, we will convert the equation of the hyperbola to standard form. Since there is a positive

x-term raised to the second power we know that this hyperbola is of the form.^{2}The given equation is almost in standard form. All we have to do is give the

x-term a denominator of^{2}as follows:1where

,h = 1,k = 2(from denominator of positive term!), anda = 1b = 3Now, we can find the requested information as follows:

Coordinates of the Center(h, k)

(1, 2)

Coordinates of the Vertices(h + a, k)and(h a, k)

and(1 + 1, 2)(11, 2)then

and(2, 2)(0, 2)

Coordinates of the Foci(h + c, k)and,(h c, k)whereSince

anda = 1, then .b = 3

and(1 + 2, 2)(12, 2)then

and(3, 2)(, 2)1