Hyberbola Conic Sections

Hyperbola The plane can intersect two nappes of the cone resulting in a 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. | d 1 – d 2 | is a constant value.

Finding An Equation Hyperbola

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). | d 1 – d 2 | is the constant. If the length of d 2 is subtracted from the left side of d1, what is the length which remains? | d 1 – d 2 | = 2a

Hyperbola - Equation Find the equation by setting the difference in the distance from the two foci equal to 2a. | d 1 – d 2 | = 2a

Hyperbola - Equation Simplify: Remove the absolute value by using + or -. Get one square root by itself and square both sides.

Hyperbola - Equation Subtract y 2 and square the binomials. Solve for the square root and square both sides.

Hyperbola - Equation Square the binomials and simplify. Get x’s and y’s together on one side.

Hyperbola - Equation Factor. Divide both sides by a 2 (c 2 – a 2 )

Hyperbola - Equation Let b 2 = c 2 – a 2 where c 2 = a 2 + b 2 If the graph is shifted over h units and up k units, the equation of the hyperbola is:

Hyperbola - Equation where c 2 = a 2 + b 2 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 units up 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. Slope = Use point-slope form y – y 1 = m(x – x 1 ) since the center is on both lines 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”.