Conics 7.3: Hyperbolas Objectives: - Analyze and graph equations of hyperbolas - Use equations to identify types of conic sections

Hyperbola: the set of all points in a plane such that the absolute value of the differences of the distances from two foci is constant. -two branches, two asymptotes -Center: midpoint of vertices and foci. -Vertices: closest point that the branches are to the center. Transverse axis: a segment that has a length of 2a units and connects the vertices. Conjugate Axis: the segment that is perpendicular to the transverse axis, passes through the center, and has length of 2b units.

The general form for a hyperbola centered at (h, k) is given below in your notes and on the next slide.

Standard Forms of Equations for Hyperbolas

Standard Forms of Equations for Hyperbolas

Steps for Graphing Hyperbolas Determine if it is horizontal or vertical. Find the center point, along with the values for a and b. Plot the center point. Use the a value to plot the two vertices. Use the b value to plot 2 additional points for drawing the guiding box and asymptotes. Draw the hyperbola.

Example 1: A) Graph the hyperbola given by .

Example 1: B) Graph the hyperbola given by

Example 2: Graph the hyperbola given by 4x2 – y2 + 24x + 4y = 28.

HW KEY: Pg. 28#1-2, 16a 1. 2. Center: (2,3) Center: (1,0) Vertices: (0,3)(4,3) Vert: (1,4)(1,-4) Foci: ( ,3) Foci: (1, ) Asymptotes: Asymptotes: 16a)

You can determine the equation for a hyperbola if you are given characteristics that provide sufficient information. Steps to write an equation: Determine if the hyperbola is horizontal or vertical. (sketch the given information) Decide which equation to use. Find the values of h, k, a, b *If given information about foci, use c2 = a2 + b2

Example 3: A) Write an equation for the hyperbola with foci (1, –5) and (1, 1) and transverse axis length of 4 units.

Example 3: B) Write an equation for the hyperbola with vertices (–3, 10) and (–3, –2) and conjugate axis length of 6 units.

Transverse Axis: 10 units PG. 28 #7: Center: (-7, 2); Asymptotes: Transverse Axis: 10 units

HW KEY: Pg. 28#3-8 3. 4. 5. 6. 7. 8.

Warm-Up: Rewrite in standard form:

You can determine the type of conic when the equation for the conic is in general form, Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. The discriminant, or B2– 4AC, can be used to identify the conic.

Example 5:Use the discriminant to identify the conic section in the equation 2x2 + y2 – 2x + 5xy + 12 = 0. Example 8:Use the discriminant to identify the conic section in the equation 4x2 + 4y2 – 4x + 8 = 0. Example 9: Use the discriminant to identify the conic section in the equation 2x2 + 2y2 – 6y + 4xy – 10 = 0.

HW KEY: Pg. 29 #11-14 11. Discriminant = -39 12 HW KEY: Pg. 29 #11-14 11. Discriminant = -39 12. Discriminant = 256 -39 < 0, B≠0 256 > 0 ELLIPSE HYPERBOLA 13. Discriminant = 0 14. Discriminant = -32 PARABOLA -32 < -, B = 0, but A ≠B ELLIPSE

Example 10: LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F than it is from station E. Find the equation for the hyperbola on which the ship is located.

Example 11: LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F than it is from station E. Find the exact coordinates of the ship if it is 125 miles from the shore.

Another characteristic that can be used to describe a hyperbola is the eccentricity. The formula for eccentricity is the same for all conics. Recall, that for an ellipse, the eccentricity is greater than 0 and less than 1. For a hyperbola, the eccentricity will always be greater than 1.

Example 6: Find the eccentricity of

HW KEY: Pg. 155 #9,16 9. e =2.09 16 a) b) c) e = 1.27