1. Conic Sections - Hyperbolas

2. 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. P F1 F2

3. Hyperbola
Asymptotes
Transverse Axis F1 F2
Vertices = (a, 0)

4. Hyperbola - Equation
For a hyperbola with a horizontal transverse axis, the standard form of the equation is: P F1 F2

5. Hyperbola F1
Transverse Axis F2

6. Hyperbola - Equation
For a hyperbola with a vertical transverse axis, the standard form of the equation is: F2 F1

7. 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.html
Vertical_Hyperbola.html

8. Find the Foci Find the foci for a hyperbola: 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
Foci are

9. Find the Foci Find the foci for a hyperbola: 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
Foci are

10. Write the Equation Write the equation of the hyperbola with foci at
(5, 0) and vertices at (3, 0) c a
From the info, it’s a horizontal transversal.
We need to find b

11. Write the Equation Write the equation of the hyperbola with foci at
(0, 13) and vertices at (0, 5) c b
From the info, it’s a vertical transversal.
We need to find a

12. Assignment