WORKSHEET KEY 12/9/2018 8:46 PM 9.5: Hyperbolas

WORKSHEET KEY 12/9/2018 8:46 PM 12/9/2018 8:46 PM 9.5: Hyperbolas 2 2

12/9/2018 8:46 PM 9.5: Hyperbolas

Hyperbolas: Conic Section Revised ©2013, vdang@houstonisd.org Hyperbolas: Conic Section 12/9/2018 8:46 PM 9.5: Hyperbolas

Hyperbola from a Cone 12/9/2018 8:46 PM 9.5: Hyperbolas

Real-Life Examples 12/9/2018 8:46 PM 9.5: Hyperbolas

Definitions Hyperbola: Set of points whose differences of the distances from any point to the foci is constant. It has two disconnected branches. Foci: Segment point joining the vertices given at a point. It is always with the TRANSVERSE axis Transverse Axis: The line segment joining the vertices. Conjugate Axis: The minor line segment joining the vertices perpendicular to the transverse axis. Asymptote: Line that a graph approaches but does not ever intersect. If the lines intersect, it becomes undefined. Eccentricity: Ratio to describe the shape of the conic e > 1 12/9/2018 8:46 PM 9.5: Hyperbolas

FORMULAS TO KNOW If the transverse axis is on the x-axis (horizontal) Horizontal Axis Standard Form: If the transverse axis is on the x-axis (horizontal) If the transverse axis is on the x-axis (horizontal) a is associated with transverse axis because it comes first. Foci of the transverse axis: Asymptotes: 12/9/2018 8:46 PM 9.5: Hyperbolas

FORMULAS TO KNOW If the transverse axis is on the y-axis (vertical) Vertical Axis Standard Form: If the transverse axis is on the y-axis (vertical) If the transverse axis is on the y-axis (vertical) a is associated with transverse axis because it comes first. Foci of the transverse axis: Asymptotes: 12/9/2018 8:46 PM 9.5: Hyperbolas

All Standard Form Equations FORMULAS TO KNOW All Standard Form Equations Center Length of Transverse Axis Length of Conjugate Axis Foci Equation Length of Latus Rectum Eccentricity 12/9/2018 8:46 PM 9.5: Hyperbolas

Horizontal Hyperbola (0, b) (–a, 0) (a, 0) F (–c, 0) F (c, 0) (h, k) (0, –b) Center: (h, k) Foci: (+c, 0) Length of Transverse Axis: 2a Vertices: (+a, 0) Length of Conjugate Axis: 2b Co-Vertices: (0, +b) Asymptotes: y – k = +b/a(x – h) Length of Latus Rectum: 2b2/a Latus Rectum: (+c, +b2/a) Eccentricity: c/a 12/9/2018 8:46 PM 9.5: Hyperbolas

Vertical Hyperbola F (c, 0) (0, –a) (–b, 0) (b, 0) (h, k) (0, a) Center: (h, k) Foci: (0, +c) Length of Transverse Axis: 2a Vertices: (0, +a) Length of Conjugate Axis: 2b Co-Vertices: (+b, 0) Asymptotes: y – k = +a/b(x – h) Length of Latus Rectum: 2b2/a Latus Rectum: (+b2/a, +c) Eccentricity: c/a F (–c, 0) 12/9/2018 8:46 PM 9.5: Hyperbolas

Ellipses vs. hyperbolas 12/9/2018 8:46 PM 9.5: Hyperbolas

When in Trouble… PLOT and GRAPH 12/9/2018 8:46 PM 9.5: Hyperbolas

Remember… SIZE DOESN’T MATTER in hyperbolas. SIZE DOES MATTER in Ellipses. WHATEVER COMES FIRST is where the axis lies. 12/9/2018 8:46 PM 9.5: Hyperbolas

Example 1 Graph F F Type: Center: Vertices: Co-Vertices: Asymptotes: B: C: Type: Center: Vertices: Co-Vertices: Asymptotes: Foci: Latus Rectum: Length of Transverse Axis: Length of Conjugate Axis: Length of Latus Rectum: Eccentricity: F F 12/9/2018 8:46 PM 9.5: Hyperbolas

Example 2 Graph F F Type: Center: Vertices: Co-Vertices: Asymptotes: B: C: Type: Center: Vertices: Co-Vertices: Asymptotes: Foci: Latus Rectum: Length of Transverse Axis: Length of Conjugate Axis: Length of Latus Rectum: Eccentricity: F F 12/9/2018 8:46 PM 9.5: Hyperbolas

YOUR TURN Graph 49y2 – 36x2 = 1764 F F Type: Center: Vertices: B: C: Type: Center: Vertices: Co-Vertices: Asymptotes: Foci: Latus Rectum: Length of Transverse Axis: Length of Conjugate Axis: Length of Latus Rectum: Eccentricity: F F 12/9/2018 8:46 PM 9.5: Hyperbolas

Example 3 Write an equation of a hyperbola where the foci is (+3, 0) and the vertices are (+2, 0). F F 12/9/2018 8:46 PM 9.5: Hyperbolas

Example 4 Write an equation where vertices are at (0, +6), asymptote is y= –6/7x and graph. 12/9/2018 8:46 PM 9.5: Hyperbolas

YOUR TURN Write an equation where center is (0, 0) opens vertically. Points used to make the rectangle are (0, +4) and (+6, 0). 12/9/2018 8:46 PM 9.5: Hyperbolas

Example 5 Graph F F Type: Center: Vertices: Co-Vertices: Asymptotes: B: C: Type: Center: Vertices: Co-Vertices: Asymptotes: Foci: Latus Rectum: Length of Transverse Axis: Length of Conjugate Axis: Length of Latus Rectum: Eccentricity: F F 12/9/2018 8:46 PM 9.5: Hyperbolas

Example 6 Graph F F Type: Center: Vertices: Co-Vertices: Asymptotes: B: C: Type: Center: Vertices: Co-Vertices: Asymptotes: Foci: Latus Rectum: Length of Transverse Axis: Length of Conjugate Axis: Length of Latus Rectum: Eccentricity: F F 12/9/2018 8:46 PM 9.5: Hyperbolas

YOUR TURN Graph F F Type: Center: Vertices: Co-Vertices: Asymptotes: B: C: Type: Center: Vertices: Co-Vertices: Asymptotes: Foci: Latus Rectum: Length of Transverse Axis: Length of Conjugate Axis: Length of Latus Rectum: Eccentricity: F F 12/9/2018 8:46 PM 9.5: Hyperbolas

Example 7 Write an equation where the foci coordinates are (6, 2) & (–8, 2) and vertices are (2, 2) & (–4, 2). F F 12/9/2018 8:46 PM 9.5: Hyperbolas

Example 8 Write an equation where center is (–3, 7), foci coordinates are (–3 + √85, 7) and vertices are (–5, 7) & (–1, 7). 12/9/2018 8:46 PM 9.5: Hyperbolas

YOUR TURN Write an equation where center is (1, –2), foci coordinates are (1 + √5, –2) and vertices are (3, –2) & (–1, –2). 12/9/2018 8:46 PM 9.5: Hyperbolas

CONSTANT DIFFERENCE Apply into the equation d = |PF1 – PF2| Determine the distance using the Distance Formula of the points, Simplify 12/9/2018 8:46 PM 9.5: Hyperbolas

Example 9 Find the constant difference for a hyperbola with foci F1 (–8, 0) and F2 (8, 0) and the point on the hyperbola (8, 30). 12/9/2018 8:46 PM 9.5: Hyperbolas

Example 9 Find the constant difference for a hyperbola with foci F1 (–8, 0) and F2 (8, 0) and the point on the hyperbola (8, 30). 34 30 F1 F2 12/9/2018 8:46 PM 9.5: Hyperbolas

Your Turn Find the constant difference for a hyperbola with foci F1 (0, –10) and F2 (0, 10) and the point on the hyperbola (6, 7.5). 12/9/2018 8:46 PM 9.5: Hyperbolas

ASSIGNMENT Worksheet 12/9/2018 8:46 PM 9.5: Hyperbolas