PARABOLA (horizontal)

Center C(h,k)

Vertices V(h±a,k)

Foci F(h±c,k)

Endpoints of the minor axis B(h,k±b)

Directrices x = h±a/e

( ) ( )1

b

ky

a

hx2

2

2

2=−+−

1

Center C(h,k)

Vertices V(h,k±a)

Foci F(h,k±c)

Endpoints of the minor axis B(h±b,k)

Directrices y = k±a/e

( ) ( )1

b

hx

a

ky2

2

2

2=−+−

2

PARABOLA (vertical)

1.4 Hyperbola

MATH 36

directrix

focus

P1

P2

F

Q1

Q2

Given the eccentricity e of a conic section, the conic is

parabola if e = 1;

ellipse if 0 < e < 1;

hyperbola if e > 1.

PQ

principal axis

vertex

Non-degenerate Conic

Objectives: At the end of this section students should be able to:

• give the standard equation of a hyperbola;

• identify parts of a hyperbola;

• sketch the graph of a hyperbola.

Hyperbola

1b

y

a

x2

2

2

2=−Standard Equation

( )0,a( )0,a−

( )b,0

( )b,0 −

centervertex

auxiliary rectangle

The line segment joining the vertices is called the transverse axis.

The line segment joining the points (0,b) and (0,-b) is called the conjugate axis.

vertex

focus( )0,c( )0,c−

focus

y=(b/a)xy=(-b/a)x

eccentricity: e = c/a

directrices:c

a

e

ax

2±=±=

222 bac +=

1b

x

a

y2

2

2

2=−Standard Equation

( )0,b( )0,b−

( )a,0

( )a,0 −

y=(a/b)xy=(-a/b)x

eccentricity: e = c/a

c

a

e

ay

2±=±=directrices:

Example 1. Given the hyperbola

14

y

9

x 22=−

Determine the (a) center, (b) principal axis, (c) vertices, (d) endpoints of the conjugate axis, (e) foci, (f) eccentricity, and (g) equations of the directrices. Sketch also the graph.

SOLUTION

center:

principal axis: x -axis

vertices:

(3,0) and (-3,0).

endpoints of conjugate axis: (0,2) and (0,-2)

3a =

2 2

19 4

x y− =

2 2 13c a b= + =2b =

(0,0)

SOLUTION

foci:

eccentricity:

equation of the directrices:

( )13 0,±

13

2

2ax

c= ±

9

13x = ±

3a = 2b =2 2 13c a b= + =

2 2

19 4

x y− =

Example 2. Given the hyperbola

19

x

16

y 22

=−

Determine the (a) center, (b) principal axis, (c) vertices, (d) endpoints of the conjugate axis, (e) foci, (f) eccentricity, and (g) equations of the directrices. Sketch also the graph.Tr

y Thiz

z on

your

own!!!

If the center of the hyperbola is at (h,k) and the line y = k as principal axis then the standard equation is given by

( ) ( ).1

b

ky

a

hx2

2

2

2=−−−

Hyperbola with center (h,k)

If the center of the hyperbola is at (h,k) and the line x = h as principal axis then the standard equation is given by

( ) ( ).1

b

hx

a

ky2

2

2

2=−−−

Hyperbola with center (h,k)

Example 3. Given the hyperbola

( ) ( )1

16

2x

4

1y 22=+−−

determine the (a) center, (b) principal axis, (c) vertices, (d) endpoints of the conjugate axis, (e) foci, (f) eccentricity, and (g) equations of the directrices. Sketch also the graph.

SOLUTION

center: (-2,1)

principal axis: x = -2

vertices:

(-2,-1) and (-2,3).

endpoints of conjugate axis: (2,1) and (-6,1)

2a =

( ) ( )1

16

2x

4

1y 22=+−−

52bac 22 =+=4b =

( )1,2( )1,6−

( )3,2−

( )1,2 −−

(-2,1)

SOLUTION

2a =

( ) ( )1

16

2x

4

1y 22=+−−

52bac 22 =+=4b =

foci:

eccentricity:

equation of the directrices:

( )521,2 ±−

5

c

a1y

2±=

52

41y ±=

(-2,1) ( )1,2( )1,6−

( )3,2−

( )1,2 −−

Example 4. Given the hyperbola

determine the (a) center, (b) principal axis, (c) vertices, (d) endpoints of the conjugate axis, (e) foci, (f) eccentricity, and (g) equations of the directrices. Sketch also the graph.

( ) ( )1

4

1

4

2 22

=+

−− yx

Try T

hizz o

n

your

own!!!

Example 5. Determine the standard equation of the hyperbola with center at ( 1,1), principal axis is vertical, length of the transverse axis and conjugate axis are 4 and 8, respectively.

SOLUTION:

center: (1, 1)

vertical principal axis

h = 1 and k = 1.

( ) ( )2 2

2 2

y k x h1

a b

− −− =

Example 5. Determine the standard equation of the hyperbola with center at ( 1,1), principal axis is vertical, length of the transverse axis and conjugate axis are 4 and 8, respectively.

SOLUTION:

length of the transverse axis: 4

b = 4

a = 2

length of the conjugate axis: 8

Example 5. Determine the standard equation of the hyperbola with center at ( 1,1), principal axis is vertical, length of the transverse axis and conjugate axis are 4 and 8, respectively.

SOLUTION:

( ) ( )2 2y 1 x 1

14 16

− −− =

standard equation

Example 6. Determine the standard equation of the hyperbola with a vertex at ( 2,0), center at the origin and its eccentricity is 3/2.

Try T

hizz o

n

your

own!!!

CENTER C(h,k)

VERTICES V(h±a,k)

FOCI F(h±c,k)

ENDPTS OF CONJUGATE AXIS B(h,k±b)

DIRECTRICES x=h±a/e

( ) ( )1

b

ky

a

hx2

2

2

2

=−−−

HYPERBOLA (horizontal)

CENTER C(h,k)

VERTICES V(h,k±a)

FOCI F(h,k±c)

ENDPTS OF CONJUGATE AXIS B(h±b,k)

DIRECTRICES y=k±a/e

( ) ( )1

b

hx

a

ky2

2

2

2

=−−−

END

HYPERBOLA (vertical)