Hyperbola ppt.
- 1. Hyperbola – a set of points in a plane whose difference of the distances from two fixed points is a constant. The Hyperbola
- 2. The Hyperbola Q 𝑑 𝐹1, 𝑄 − 𝑑 𝐹2, 𝑄 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = ±2𝑎 Hyperbola – a set of points in a plane whose difference of the distances from two fixed points is a constant. 𝑑 𝐹1, 𝑃 − 𝑑 𝐹2, 𝑃 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = ±2𝑎
- 3. The Hyperbola Foci – the two fixed points, 𝐹1 𝑎𝑛𝑑 𝐹2, whose difference of the distances from a single point on the hyperbola is a constant. Transverse axis – the line that contains the foci and goes through the center of the hyperbola. Vertices – the two points of intersection of the hyperbola and the transverse axis, 𝑉1 𝑎𝑛𝑑 𝑉2 . Conjugate axis – the line that is perpendicular to the transverse axis and goes through the center of the hyperbola. Conjugate axis Center – the midpoint of the line segment between the two foci. Center
- 7. The Hyperbola (−𝑏, 0) (𝑏, 0)
- 9. The Hyperbola Identify the direction of opening, the coordinates of the center, the vertices, and the foci. Find the equations of the asymptotes and sketch the graph. 𝑥2 16 − 𝑦2 9 = 1 Vertices of transverse axis: 𝑎2 = 16 Center: (0,0) Equations of the Asymptotes Foci 𝑏2 = 9 𝑏2 = 𝑐2 − 𝑎2 𝑎 = ±4 −4,0 𝑎𝑛𝑑 (4,0) 𝑏 = ±3 0,3 𝑎𝑛𝑑 (0, −3) 9 = 𝑐2 − 16 𝑐2 = 25 𝑐 = ±5 −5,0 𝑎𝑛𝑑 (5,0) 𝑦 − 𝑦1 = ± 𝑏 𝑎 (𝑥 − 𝑥1) 𝑦 − 0 = ± 3 4 (𝑥 − 0) 𝑦 = ± 3 4 𝑥
- 10. The Hyperbola 𝑦2 4 − 𝑥2 16 = 1 Vertices of transverse axis: 𝑎2 = 4 Center: (0,0) Equations of the Asymptotes Foci 𝑏2 = 16 𝑏2 = 𝑐2 − 𝑎2 𝑎 = ±2 0, −2 𝑎𝑛𝑑 (0,2) 𝑏 = ±4 −4,0 𝑎𝑛𝑑 (4,0) 16 = 𝑐2 − 4 𝑐2 = 20 𝑐 = ±2 5 0, −2 5 𝑎𝑛𝑑 (0,2 5) 𝑦 − 𝑦1 = ± 𝑎 𝑏 (𝑥 − 𝑥1) 𝑦 − 0 = ± 2 4 (𝑥 − 0) 𝑦 = ± 1 2 𝑥 Identify the direction of opening, the coordinates of the center, the vertices, and the foci. Find the equations of the asymptotes and sketch the graph.
- 11. The Hyperbola (𝑦 − 𝑘)2 𝑎2 − (𝑥 − ℎ)2 𝑏2 = 1 Find b: 𝑎2 = 1 Center: 𝑏2 = 9 − 1 = 8 𝑏2 = 𝑐2 − 𝑎2 𝑎 = 4 − 3 = 1 Equation of the Hyperbola 𝑏 = ±2 2 −4 − 2 2, 0 𝑎𝑛𝑑 (−4 + 2 2, 0) 𝑐2 = 9 𝑐 = 3 − 0 = 3 −6.83,3 𝑎𝑛𝑑 (−1.17,3) A hyperbola has a focus at (−4,0) and vertices at −4,4 𝑎𝑛𝑑 (−4,2). What is its equation? Graph the hyperbola. −4 + (−4) 2 , 4 + 2 2 (−4,3) (𝑦 − 3)2 1 − (𝑥 + 4)2 8 = 1
- 12. The Hyperbola Center: Equations of the Asymptotes 𝑎 = 1 𝑏 = ±2 2 𝑦 − 𝑦1 = ± 𝑎 𝑏 (𝑥 − 𝑥1) 𝑦 − 3 = ± 1 2 2 (𝑥 + 4) A hyperbola has a focus at (−4,0) and vertices at −4,4 𝑎𝑛𝑑 (−4,2). What is its equation? Graph the hyperbola. (−4,3) 𝑦 − 3 = ± 2 4 (𝑥 + 4)
- 13. The Hyperbola Find the center, the vertices of the transverse axis, the foci and the equations of the asymptotes using the following equation of a hyperbola. 𝑦2 − 4𝑥2 − 72𝑥 + 10𝑦 − 399 = 0 𝑦2 + 10𝑦 − 4(𝑥2 + 18𝑥) = 399 10 2 = 5 52 = 25 18 2 = 9 92 = 81 𝑦2 + 10𝑦 + 25 − 4(𝑥2+18𝑥 + 81) = 399 + 25 − 324 (𝑦 + 5)2−4(𝑥 + 9)2= 100 (𝑦 + 5)2 100 − 4(𝑥 + 9)2 100 = 1 (𝑦 + 5)2 100 − (𝑥 + 9)2 25 = 1 𝑦2 + 10𝑦 − 4𝑥2 − 72𝑥 = 399 Opening up/down
- 14. The Hyperbola Find the center, the vertices of the transverse axis, the foci and the equations of the asymptotes using the following equation of a hyperbola. 𝑦2 − 4𝑥2 − 72𝑥 + 10𝑦 − 399 = 0 (𝑦 + 5)2 100 − (𝑥 + 9)2 25 = 1 Center: (−9, −5) Vertices: 𝑎2 = 100 𝑎 = 10 −9, −5 − 10 𝑎𝑛𝑑 (−9, −5 + 10) −9, −15 𝑎𝑛𝑑 (−9,5) Foci: 25 = 𝑐2 − 100 𝑏2 = 𝑐2 − 𝑎2 −9, −5 − 5 5 𝑎𝑛𝑑 (−9, −5 + 5 5) 𝑐 = 125 = 5 5 −9, −16.18 𝑎𝑛𝑑 (−9,6.18) 𝑐2 = 125
- 15. The Hyperbola Find the center, the vertices of the transverse axis, the foci and the equations of the asymptotes using the following equation of a hyperbola. 𝑦2 − 4𝑥2 − 72𝑥 + 10𝑦 − 399 = 0 (𝑦 + 5)2 100 − (𝑥 + 9)2 25 = 1 Center: (−9, −5) 𝑎 = 10 Equations of the Asymptotes 𝑦 − 𝑦1 = ± 𝑎 𝑏 (𝑥 − 𝑥1) 𝑏 = 5 𝑦 − (−5) = ± 10 5 (𝑥 − (−9)) 𝑦 + 5 = ±2 (𝑥 + 9)