conic sections hyperbolas

Colleen BeaudoinFebruary, 2009

Review: The geometric definition relies on a cone and a plane intersecting it

Algebraic definition: a set of points in the plane such that the difference of the distances from two fixed points, called foci, remains constant.

x

y

From each point in the plane, the difference of the distances to the foci is a constant. Example:f1

f2

d1d2

fociPoint A: d1-d2 = c Point B: d1-d2 = c

B

A

d1

d2

x

y

f1 f2

foci

Center

Transverse Axis

Conjugate Axis

Vertices

Algebraic Definition of a hyperbola: a set of points in the plane such that the difference of the distances from two fixed points, called foci, remains constant.

How is the definition similar to that of an ellipse?

How is it different?

Both variables are squared. Equation:

Compare the equations of ellipses and hyperbolas.

What makes the hyperbola different from the parabola?

What makes the hyperbola different from a circle?

Procedure to graph:1. Put in standard form (above): x squared

term - y squared term = 12. Determine if the hyperbola is opening

vertically or horizontally. (If x is first, it’s horizontal. If y is first, it’s vertical.)

3. Plot the center (h,k)4. Plot the endpoints of the horizontal axis by

moving “a” units left and right from the center.

2 2 2 2

2 2 2 2

( - ) ( - ) ( - ) ( - )1 or x h y k y k x ha b b a

To graph:5. Plot the endpoints of the vertical axis by

moving “b” units up and down from the center.

Note: The line segment that contains the vertices of the hyperbola is known as the transverse axis. The other axis is the conjugate axis.

6. Draw a rectangle such that each of the axis endpoints is the midpoint of a side.

2 2 2 2

2 2 2 2

( - ) ( - ) ( - ) ( - )1 or x h y k y k x ha b b a

To graph:7. Sketch the diagonals of the rectangle

and extend them outside of the rectangle. (These are the asymptotes of the hyperbola.)

8. Draw each branch of the hyperbola – Be sure to go through the vertex of each (the endpoint of the transverse axis) and approach the asymptotes.

2 2 2 2

2 2 2 2

( - ) ( - ) ( - ) ( - )1 or x h y k y k x ha b b a

To graph:9. Use the following formula to help locate the

foci: c2 = a2 + b2

Move “c” units left and right form the center if the transverse axis is horizontal

OR Move “c” units up and down form the center if the transverse axis is vertical

Label the points f1 and f2 for the two foci.

Note: It is not necessary to plot the foci to graph the hyperbola, but it is common practice to locate them.

2 2 2 2

2 2 2 2

( - ) ( - ) ( - ) ( - )1 or x h y k y k x ha b b a

The equation of each asymptote can be found by using the point-slope formula. Use the center as “the point” and slope can be found by counting on the graph (from the point to the corner of the rectangle).

Or the following formulas can be used:With horizontal transverse axis: b by = k + (x - h) and y = k - (x - h)a a

With vertical transverse axis: a ay = k + (x - h) and y = k - (x - h)b b

1. Put in standard form. Done2. Determine if the hyperbola is opening vertically

or horizontally. Vertically because “y” is first.3. Identify the center. (0,0)4. Identify the endpoints of the horizontal axis. (6,0) and (-6,0)5. Identify the endpoints of the vertical axis. (0,8) and (0,-8) Which pair of endpoints are the vertices? (0,8) and (0,-8)

2 2

164 36y x

6. Draw a rectangle such that each of the axis endpoints is the midpoint of a side.

7. Sketch the asymptotes of the hyperbola.8. Draw each branch of the hyperbola – Be

sure to go through the vertex of each (the endpoint of the transverse axis) and approach the asymptotes.

2 2

164 36y x

9. Locate the foci. (0,10) and (0,-10) 10. Find the equations of the asymptotes.

2 2

164 36y x

4 4 and 3 3

y x y x

x

y

Transverse axis

Conjugate axis

Center

Asymptotes

2 2

164 36y x

2 2( 3) ( 2) 136 16x y

1. Put in standard form. Done2. Determine if the hyperbola is opening vertically

or horizontally. Horizontally because “x” is first.3. Identify the center. (3,-2)4. Identify the endpoints of the horizontal axis. (-3,-2) and (9,-2)5. Identify the endpoints of the vertical axis. (3,2) and (3,-6) Which pair of endpoints are the vertices? (-3,-2) and (9,-2)

6. Draw a rectangle such that each of the axis endpoints is the midpoint of a side.

7. Sketch the asymptotes of the hyperbola.8. Draw each branch of the hyperbola – Be

sure to go through the vertex of each (the endpoint of the transverse axis) and approach the asymptotes.

2 2( 3) ( 2) 136 16x y

9. Locate the foci. (3+2√13,-2) and (3-2√13,-2)10. Find the equations of the asymptotes. 4 4 and

3 3y x y x

2 2( 3) ( 2) 136 16x y

2 2( 3) ( 2) 136 16x y

x

y

Exp. 3: Write the equation in Exp. 3: Write the equation in standard form and graph:standard form and graph:2 24 64y x

2 2

164 16y x

1. Put in standard form.

2 2

164 16y x

x

y

Write the equation for a hyperbola with x-intercepts at 5 and -5 and foci (6,0) and (-6,0).

1) How can you tell if the graph of an equation will be a line, parabola, circle, ellipse, or hyperbola?

2) What’s the standard form of a hyperbola?

3) What’s the standard form of an ellipse?4) What’s the standard form of a parabola?5) What’s the standard form of a circle?6) How are the various equations similar

and different?