HyperbolasLESSON 10.5

Objective

Define hyperbolas and parts of a hyperbola

Graph hyperbolas with center at the origin

Find the equation of a hyperbola

Find the asymptotes of a hyperbola at the origin

Define Hyperbolas

A hyperbola is the set of all points on a plane whose distance from

two fixed focal points (foci) subtract to a constant number.

NOTE: an ellipse adds to be a constant

distance from the foci while a hyperbola

subtracts to be a constant distance.

Define Hyperbolas

The transverse axis is the line containing the foci, the midpoint

of which is the center. The two points of intersection of the

hyperbola and the transverse

axis are the vertices (𝑉1, 𝑉2).

Define Hyperbolas

With a center at the origin, equations for the two

types of hyperbolas follow:

Horizontal Vertical

𝑥2

𝑎2−

𝑦2

𝑏2= 1 or

𝑦2

𝑎2−

𝑥2

𝑏2= 1

Graph Hyperbolas

To graph hyperbolas

1. Identify the center, 𝑎, and 𝑏.

2. Draw a box with 𝑎 being the distance from the

center along the transverse axis

3. Connect opposite corners to create asymptotes

4. Draw hyperbola with respect to asymptotes

Graph Hyperbolas

Identify 𝑎, 𝑏, and the transverse axis. Then graph.

1) 𝑦2

4−

𝑥2

16= 1

Graph Hyperbolas

Identify 𝑎, 𝑏, and the transverse axis. Then graph.

2) 𝑥2

16−

𝑦2

9= 1

Graph Hyperbolas

Identify 𝑎, 𝑏, and the transverse axis. Then graph.

3) 𝑥2 −𝑦2

25= 1

Graph Hyperbolas

To find the vertices of hyperbolas

1. Identify the center, 𝑎, and the transverse axis

2. ±𝑎 to the center along the transverse axis

Graph Hyperbolas

Find the center, 𝑎, and both vertices (𝑉1, 𝑉2)

4) 𝑦2

4−

𝑥2

16= 1 5)

𝑥2

16−

𝑦2

9= 1

6) 𝑥2 −𝑦2

25= 1

Graph Hyperbolas

The focal points (foci) are a set distance 𝒄 away

from the center along the transverse axis.

𝑐2 = 𝑎2 + 𝑏2

Graph Hyperbolas

Find the center, 𝑐, and both foci (𝐹1, 𝐹2)

7) 𝑦2

4−

𝑥2

16= 1 8)

𝑥2

16−

𝑦2

9= 1

Equation of Hyperbolas

Find the equation of the hyperbola with the

following information.

9) Vertices (±3,0)

Foci (±5,0)

Equation of Hyperbolas

Find the equation of the hyperbola with the

following information.

10) Vertices (0, ±12)

Foci (0, ±13)