Transcript Notes
Today in Precalculus • Go over homework • Notes: Hyperbolas Completing the square Eccentricity • Homework Hyperbolas Prove that the graph of 9x2 – 4y2 – 18x + 8y + 41 = 0 is an hyperbola. Find the center, vertices and foci. Then graph the hyperbola by hand. 9x2 – 18x – 4y2 + 8y = -41 9(x2 – 2x) – 4(y2 – 2y) = -41 9(x2 – 2x + 1) – 4(y2 – 2y + 1) = -41 + 9 - 4 9(x – 1)2 – 4(y – 1)2 = -36 9( x 1) 2 4( y 1) 2 36 36 36 36 ( y 1) 2 ( x 1) 2 1 9 4 ( y 1) ( x 1) 1 9 4 Center: (1, 1) a = ±3 Vertices: (1, -2), (1, 4) c2 = 9 +4 = 13 c = ±3.6 Foci: (1, -2.6), (1,4.6 ) Pts on conjugate axis b = ±2 (-1,1), (3,1) 2 y yy 2 x x 3 Asymptotes : y ( x 1) 1 2 x Example 2 Prove that the graph of 9y2 – 25x2 + 72y – 100x +269 = 0 is an hyperbola. Find the center, vertices and foci. Then graph the hyperbola by hand. 9y2 + 72y – 25x2 – 100x = -269 9(y2 + 8y) – 25(x2 + 4x) = - 269 9(y2 + 8y + 16) – 25(x2 + 4x + 4) = -269 +144 -100 9(y + 4)2 – 25(x + 2)2 = -225 9( y 4) 2 25( x 2) 2 225 225 225 225 ( x 2)2 ( y 4)2 1 9 25 ( x 2) ( y 4) 1 9 25 2 2 Center: (-2, -4) a=±3 Vertices: (-5, -4), (1, -4) c2 = 9 + 25 =34 c = ±5.8 Foci: (-7.8, -4), (3.8, -4) Pts on conjugate axis b = ±5 5 (-2, -9), (-2, 1) Asymptotes : y ( x 2) 4 3 Eccentricity c a2 b 2 e a a Where a is the semitranvserse axis and b is the semiconjugate axis. Example a: 9 49 e Example b: 3 2.539 16 25 e 1.601 4 Homework Page 664: 47-50 Ellipse and Hyperbola Quiz: Tuesday, February 26