Name: ________________________ Per: _______ Date: ___________ Geometry Honors Conics: Hyperbola A. Slicing the cone (conic section) A hyperbola is formed by slicing a cone ______________________ B. The locus definition A hyperbola is a locus of points whose difference of distances from two given foci is constant. C. The standard form of hyperbola centered at the origin. x2 y 2 y2 x 2 1 or 1 a2 b2 a2 b2 Where a b or b a or a b The length of the transverse axis is 2 a The length of the conjugate axis is 2 b The foci are a distance c from the origin, where c2 a2 b2 b a The lines y x or y x are the asymptotes for the hyperbola. These asymptotes serve as excellent a b guides for sketching the graph. 1 Name: ________________________ Per: _______ Date: ___________ Geometry Honors D. Important parts of the hyperbola (terminology) Focus (or foci): Points that are used to generate the hyperbola. The foci of the hyperbola are not ON the hyperbola, but are always found “inside” the boomerang-looking pieces. Center: Midpoint of the foci Asymptotes: Lines that the hyperbola approaches but never reaches. They help “frame” the hyperbola. Transverse Axis: The line segment the vertices are called the transverse axis. The foci lie beyond the endpoints of the transverse axis. Conjugate axis: The graph does not cross the other axis. Its endpoints are not points on the hyperbola, however, are very important in creating an auxiliary rectangle that assists in sketching the graph. x 2 y2 For a hyperbola with a horizontal transverse axis: 1 a2 b 2 b y x is the asymptotes for the hyperbola. a y 2 x2 For a hyperbola with a vertical transverse axis: 1 a2 b 2 a y x is the asymptotes for the hyperbola. b y Remember slope = x 2 Name: ________________________ Per: _______ Date: ___________ Geometry Honors 1-2. Sketch each of the following hyperbolae by doing the following: a) Graph and label the vertices. b) Determine, graph, and label the asymptotes. c) Draw the hyperbola. d) Determine the c-value and the coordinates of the foci. e) Graph and label the foci. x 2 y2 a. 1 b. 4y2 x 2 4 4 25 3 Name: ________________________ Per: _______ Date: ___________ Geometry Honors 3. Find the equation of the hyperbola with the center at the origin, one focus at (3, 0) and one vertex at (-2, 0). Graph the equation. 4 Name: ________________________ Per: _______ Date: ___________ Geometry Honors SUMMARY All hyperbolas are centered at the origin and are of the following forms: Helpful hints: If the “x2” appears first in the equation, then the intercepts (vertices) lie on the x-axis and so do the foci. If the “y2” appears first in the equation, then the intercepts (vertices) lie on the y-axis and so do the foci. The asymptotes pass through the origin. Whatever is on top of the fraction is the number that is “with” y, just like when writing the slope of a line the “change in y” goes on top. Summary Table Equation A B Opens Vertices Coordinates Asymptotes up/down or (only two!!) of the Foci Y = … left/right? 1. 2. 3. 4. 5. 6. 5 Name: ________________________ Per: _______ Date: ___________ Geometry Honors Homework 6 Name: ________________________ Per: _______ Date: ___________ Geometry Honors 7 Name: ________________________ Per: _______ Date: ___________ Geometry Honors 8 .