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8.3 Hyperbolas
Geometry of a Hyperbola
Transverse Axis (focal axis)
Hyperbola with Center (h, k)
Standard Equation
Opens
Focus
Center
( x  h) 2 ( y  k ) 2

1
a2
b2
Opens Left and Right
Conjugate Axis
( y  k ) 2 ( x  h) 2

1
a2
b2
Opens Up and Down
Vertices
Pythagorean Relation
Directions: Draw the diagram of the hyperbola with all the different parts labeled.
Example: Find the vertices and foci of the hyperbolas below; then graph it.
y2 x2
a.

1
16 7
b. 4 x 2  9 y 2  36
8.3 Hyperbolas
Example: Write the equation of the hyperbolas with the given information.
a. Center (0,0)
Foci 0,6
Vertices 0,5
b. Center (2, 1)
Vertices (5, 1) and (-1, 1)
Length of the Conjugate = 4
Eccentricity:
e = 0 for ___________________
e = 1 for _________________
0  e  1 for _______________
e > 1 for _________________
Identify the Type of Conic:
Recall general forms:
Parabola:
Ellipse:
Hyperbola:
8.3 Hyperbolas
Examples:
y2 x2
A.

1
16 49
y 2 x  6 

1
B.
36
20
Shape:_____________
Shape: ______________________
a = _______
a = _______
b = _________
2
b = _________
C. x2 – 6x – y – 3 = 0
D. x2 – y2 – 2x + 4y – 6 = 0
Shape: _______________________ Prove it!
Shape: _________________________ Prove it!
E. y2 – 6x – 4y – 13 = 0
F. 2x2 – 3y2 – 12x – 24y + 60 = 0
Shape: __________________ Prove it!
Shape: _______________________ Prove it!