Conic Sections - Hyperbolas

Section 10.4

Hyperbola

Hyperbola – is the set of points (P) in a plane such that the difference of the distances from P to two fixed points F1 and F2 is a given constant k.

F2F1kPFPF 21

P

Hyperbola

F2F1

Transverse Axis

Vertices = (a, 0)

Asymptotes

xa

by

xa

by

Hyperbola - Equation

For a hyperbola with a horizontal transverse axis, the standard form of the equation is:

F2F11

2

2

2

2

b

y

a

x

P

Hyperbola

F2

F1 Transverse Axis

xb

ay

xb

ay

Hyperbola - Equation

For a hyperbola with a vertical transverse axis, the standard form of the equation is:

F2

F1

12

2

2

2

a

x

b

y

Hyperbola

Definitions:• a – is the distance between the vertex and the center of

the hyperbola• b – is the distance between the tangent to the vertex

and where it intersects the asymptotes• c – is the distance between the foci and the center

Relationships:The distances a, b and c form a right triangle and can be

used to construct the hyperbola.Horizontal_Hyperbola.htmlVertical_Hyperbola.html

Find the Foci

Find the foci for a hyperbola: 1925

22

yx

a2 b2

From the form, we know it’s a horizontal transverse axis. We know the foci are at (c, o ) and that c2 = a2 + b2

34

925

c

Foci are 0,34

Find the Foci

Find the foci for a hyperbola: 12549

22

xy

b2 a2

From the form, we know it’s a vertical transverse axis. We know the foci are at (0, c ) and that c2 = a2 + b2

74

2549

c

Foci are 74,0

Write the Equation

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

From the info, it’s a horizontal transversal. We need to find b

c a

b

b

b

b

4

16

925

35

2

2

222

1169

22

yx

Write the Equation

Write the equation of the hyperbola with foci at (0, 13) and vertices at (0, 5)

From the info, it’s a vertical transversal. We need to find a

c b

2

2

222

144

25169

513

a

a

a

114425

22

xy

Assignment