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Hyberbola. Conic Sections. The plane can intersect two nappes of the cone resulting in a hyperbola . Hyperbola. Hyperbola - Definition. A hyperbola is the set of all points in a plane such that the difference in the distances from two points (foci) is constant.

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Hyberbola

Hyberbola

Conic Sections



Hyperbola definition
Hyperbola - Definition a

A hyperbola is the set of all points in a plane such that the difference in the distances from two points (foci) is constant.

| d1 – d2 | is a constant value.


Hyperbola definition1
Hyperbola - Definition a

What is the constant value for the difference in the distance from the two foci? Let the two foci be (c, 0) and (-c, 0). The vertices are (a, 0) and (-a, 0).

| d1 – d2| is the constant.

If the length of d2 is subtracted from the left side of d1, what is the length which remains?

| d1 – d2 | = 2a


Hyperbola equation
Hyperbola - Equation a

where c2 = a2 + b2

Recognition:How do you tell a hyperbola from an ellipse?

Answer:A hyperbola has a minus (-) between the terms while an ellipse has a plus (+).


Graph example 1

Graph - Example #1 a

Hyperbola


Hyperbola graph
Hyperbola - Graph a

Graph:

Center:

(-3, -2)

The hyperbola opens in the “x” direction because “x” is positive.

Transverse Axis:

y = -2


Hyperbola graph1
Hyperbola - Graph a

Graph:

Vertices

(2, -2) (-4, -2)

Construct a rectangle by moving 4 unitsup and down from the vertices.

Construct the diagonals of the rectangle.


Hyperbola graph2
Hyperbola - Graph a

Graph:

Draw the hyperbola touching the vertices and approaching the asymptotes.

Where are the foci?


Hyperbola graph3
Hyperbola - Graph a

Graph:

The foci are 5 units from the center on the transverse axis.

Foci: (-6, -2) (4, -2)


Hyperbola graph4
Hyperbola - Graph a

Graph:

Find the equation of the asymptote lines.

4

3

Use point-slope formy – y1 = m(x – x1) since the center is on both lines.

-4

Slope =

Asymptote Equations


Graph example 2

Graph - Example #2 a

Hyperbola


Hyperbola graph5
Hyperbola - Graph a

Sketch the graph without a grapher:

Recognition:How do you determine the type of conic section?

Answer:The squared terms have opposite signs.

Write the equation in hyperbolic form.


Hyperbola graph6
Hyperbola - Graph a

Sketch the graph without a grapher:


Hyperbola graph7
Hyperbola - Graph a

Sketch the graph without a grapher:

Center:

(-1, 2)

Transverse Axis Direction:

Up/Down

Equation:

x=-1

Vertices:

Up/Down from the center or


Hyperbola graph8
Hyperbola - Graph a

Sketch the graph without a grapher:

Plot the rectangular points and draw the asymptotes.

Sketch the hyperbola.


Hyperbola graph9
Hyperbola - Graph a

Sketch the graph without a grapher:

Plot the foci.

Foci:


Hyperbola graph10
Hyperbola - Graph a

Sketch the graph without a grapher:

Equation of the asymptotes:


Finding an equation

Finding an Equation a

Hyperbola


Hyperbola find an equation
Hyperbola – Find an Equation a

Find the equation of a hyperbola with foci at (2, 6) and (2, -4). The transverse axis length is 6.


Conic section recogition

Conic Section a Recogition


Recognizing a conic section
Recognizing a Conic Section a

Parabola -

One squared term. Solve for the term which is not squared. Complete the square on the squared term.

Ellipse -

Two squared terms. Both terms are the same “sign”.

Circle -

Two squared terms with the same coefficient.

Hyperbola -

Two squared terms with opposite “signs”.