Hyberbola
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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

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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.

| d1 – d2 | is a constant value.


Hyperbola - Definition

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

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

Hyperbola


Hyperbola - Graph

Graph:

Center:

(-3, -2)

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

Transverse Axis:

y = -2


Hyperbola - Graph

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 - Graph

Graph:

Draw the hyperbola touching the vertices and approaching the asymptotes.

Where are the foci?


Hyperbola - Graph

Graph:

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

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


Hyperbola - Graph

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

Hyperbola


Hyperbola - Graph

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 - Graph

Sketch the graph without a grapher:


Hyperbola - Graph

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 - Graph

Sketch the graph without a grapher:

Plot the rectangular points and draw the asymptotes.

Sketch the hyperbola.


Hyperbola - Graph

Sketch the graph without a grapher:

Plot the foci.

Foci:


Hyperbola - Graph

Sketch the graph without a grapher:

Equation of the asymptotes:


Finding an Equation

Hyperbola


Hyperbola – Find an Equation

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


Conic Section Recogition


Recognizing a Conic Section

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”.


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