Conics: Hyperbolas

Graph a hyperbola and identify its

important parts.

Write the equation of a hyperbola in

standard form given its important parts.

Identify a hyperbola and write the

equation in standard form given a conic

equation in non standard form.

Standard Forms of a Hyperbola









2

2

2

2
1
y

k

x

h

a

b








The standard form of the equation of an
hyperbola with center at (h, k).

Vertices 


,h k

a

Vertical

Horizontal









2

2

2

2
1
x

h

y

k

a

b








Vertices 


,

h

a k

2

2

2

:

Foci

c

a

b








,

h

c k
Foci



,h k

c
Foci

x
a
b
y




Asymptotes
a
y

x
b
 
Asymptotes

#1 Graph the hyperbola

144

16

9

2

2





y

x

1

9

16

2

2



y

x




0 ,4
Vertices

x

y
4
3


Asymptotes

4



a

3



b

9

16

c
5






0 ,5
Focus

#2 Graph the hyperbola

2

2

25

25

y

x





1

25

2

2



 x
y




5

,0 
Vertices

x

y
1
5


Asymptotes

5



a

1b

1

25

c
1.5






0,

26


Focus

26



#3 Graph the hyperbola









2

2

2

y

x



 


Asymptotes

2

a 

4

b 

16

4

c 


2 5

4.4








2

2 5, 2


Focus

x  2




2

4
y  2




2

16
 1




2, 2

Center



 


4, 2 , 0, 2




Vertices

#4 Graph the hyperbola

y  2




2

4

x  3




2

9

 1




2 ,3
Center








2
2

3
3
y

x



 



Asymptotes

2



a

3



b



 


4 ,3

,

0 ,3




Vertices

6.3

c




3,2

13




Foci

Write an equation of the hyperbola with foci
at (0, ±5) and vertices at (0, ±3).

#5 Writing Equations of Hyperbolas

1
2

2

2

2



 b

x

a
y

3



a

5

c

2

2

2

b

a

c





2

9

25

b



4

b

1
16

9

2

2



x

y

Write an equation of the hyperbola with foci
at and vertices at (4,3) and (2,3).

#6 Writing Equations of Hyperbolas

a  1 c 

10
2

2

2

b

a

c





10 1 b2

b  3

3 

10,3












2

2

2

2
1
x

h

y

k

a

b








Center: 3,3




x  3




2

1


y  3




2

9
 1

Write an equation of the hyperbola with foci at (4,5)
and (4,-3) and vertices at (4,4) and (4,-2).

#7 Writing Equations of Hyperbolas

a  3

c  4

2

2

2

b

a

c





16  9  b2

b 

7









2

2

2

2
1
y

k

x

h

a

b








Center: 4,1




y 1




2

9


x  4




2

7
 1



 









__1

__4

4

__

4

__

4

4

2

2

















y

y

x

x

#8 Write in Standard Form

0

4

4

16

4

2

2











y

x

y

x

4

4

16

4

2

2









y

y

x

x

4








1
16

2

4
2

2

2






y

x

4

4

4








16

2

2

4

2

2









y

x








4

16

4

2

2

4

2

2













y

x

16 y

2  4y  __


 9 x

2  6x  __


16116(__) 9(__)

#9 Write in Standard Form

9x216y2 54x  64y 161 0

16y2 64y  9x2 54x 161

4









2

2

2

3
1
9

16

y

x








4
9

9









2

2
16

2

9

3

144

y

x

















2

2
16

2

9

3

161 64

81

y

x











