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January 14, 2015 9.3 Hyperbolas Objective: Write the standard form of the equation of a hyperbola, and analyze and sketch the graphs of hyperbolas. Hyperbola set of all points (x, y) in a plane, the difference of whose distances from two distinct fixed points (foci) is constant. d1 Center - midpoint between the foci. d2 branches Vertices - points at which the line segment through the foci meets the hyperbola. Transverse axis - line segment joining the vertices. *Note: d2 - d1 = positive constant http://www.brightstorm.com/math/precalculus/conic-sections/the-hyperbola/#ooid=ZzaWV3NDoSM36UvSLAMq-B http://www.intmath.com/plane-analytic-geometry/6-hyperbola.php Standard form e ot ot e sy m pt pt m sy A A Hyperbola with horizontal transverse Center: (h, k) Transverse axis of length 2a Conjugate axis of length 2b Vertices: (h ± a, k) (x-h)2 (y-k)2 =1 Foci: (h ± c, k) a2 b2 Asymptote: y = k± b a (x - h) le" ang ect R gic "Ma sy m pt A pt m sy e ot NOTE: In both cases, c2 = a2 + b2. a = distance from center to the vertices b = distance from center to endpoint of conjugate axis c = distance from the center to the foci A Transverse axis of length 2a " Conjugate axis of length 2b gle tan c e Vertices: (h, k ± a) R gic "Ma Foci: (h, k ± c) Asymptote: y = k ± a (x - h) (y-k)2 (x-h)2 b =1 a2 b2 ot e Hyperbola with vertical transverse Center: (h, k) January 14, 2015 Ex1: Find the center, vertices, asymptotes, and foci of the hyperbola given by 4y2 - 9x2 = 36. a = 3, b = 2, and c2 = 9 + 4 ⇒ c ≈ 3.6. So, the center is (0, 0), the vertices are at (0, ±3), the asymptotes are y = ±(3/2)x, and the foci are at (0, ±3.6). Ex2: Find the standard form of the equation of the hyperbola centered at the origin with transverse axis of length 4 and foci at (±3, 0). We know that a = 2 and c = 3. Next b2 = 32 - 22 or b =± √5. So, the equation is x2/4 - y2/5 = 1. Ex3: Sketch the graph of the hyperbola given by 4x2 - 9y2 - 24x - 72y - 72 = 0. Eccentricity The eccentricity of a hyperbola is given by e = ca and e > 1. the larger the eccentricity, the closer the branches of the hyperbola are to being lines. January 14, 2015 Applications Orbits of comets and planets p.654 http://mathforum.org/~sanders/geometry/GP20Hyperbola.html Ex4: There is a listening station located at A(2200, 0) (in feet) and another at B(-2200, 0). An explosion is heard at station A one second before it is heard at station B. Where was the explosion located? *Hint: Speed of sound = 1100 ft/sec The explosion o c = 2200. Sinc second, 2a = 1 b = 2200 - 5 right branch of 2 Classifying Conics from its General Equation The graph of Ax + Cy + Dx + Ey + F = 0 is: 1. a circle if A = C. 2 2 2. a parabola if AC = 0. 3. an ellipse if AC > 0. 4. a hyperbola if AC < 0. Ex5: Classify each of the following. a) ellipse (AC = 20) a) 4x + 5y - 9x + 8y = 0 b) parabola (AC = 0) b) 2x - 5x + 7y - 8 = 0 c) circle (A = C) c) 7x + 7y - 9x + 8y - 16 = 0 d) hyperbola (AC = -20) d) 4x - 5y - x + 8y + 1 = 0 2 2 2 2 2 2 2 2 January 14, 2015