MATHPOWER TM 12, WESTERN EDITION Chapter 3 Conics

The hyperbola is the locus of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to two given points in the plane, the foci, is constant. F1F1 F2F2 a c O A1A1 A2A2 A 1 and A 2 are called the vertices. Line segment A 1 A 2 is called the transverse axis and has a length of 2a units. The distance from the centre to either focus is represented by c. Both the transverse axis and its perpendicular bisector are lines of symmetry of the hyperbola. a c Transverse axis The Hyperbola

F1F1 F2F2 P(x, y) | PF 1 - PF 2 | = 2a Locus Definition

The diagram shows a graph of a hyperbola with a rectangle centred at the origin. The points A 1, A 2, B 1 and B 2 are the midpoints of the sides of the rectangles. The hyperbola lies between the lines containing its diagonals. As | x | increases, the hyperbola comes closer to these lines. These lines are asymptotes. The line segment B 1 B 2 is called the conjugate axis. The conjugate axis has a length of 2b units. A1A1 A1A1 B1B1 B2B2 For a hyperbola, the value of b can be found using the Pythagorean Theorem, a 2 + b 2 = c The Hyperbola Centred at the Origin

(c, 0) (-c, 0) F1F1 F2F2 (-a, 0) (a, 0) A1A1 A2A2 B (0, b) B (0, -b) The Standard Equation of a Hyperbola With Centre (0, 0) and Foci on the x-axis The equation of a hyperbola with the centre (0, 0) and foci on the x-axis is: The length of the transverse axis is 2a. The length of the conjugate axis is 2b. The vertices are (a, 0) and (-a, 0). The foci are (c, 0) and (-c, 0). The slopes of the asymptotes are The equations of the asymptotes 3.5.5

The Standard Equation of a Hyperbola with Centre (0, 0) and Foci on the y-axis [cont’d] F 1 (0, c) F 2 (0, -c) A 1 (0, a) A 2 (0, -a) B 2 (b, 0) B 1 (-b, 0) The equation of a hyperbola with the centre (0, 0) and foci on the y-axis is: The length of the transverse axis is 2a. The length of the conjugate axis is 2b. The vertices are (0, a) and ( 0, -a). The foci are (0, c) and (0, -c). The slopes of the asymptotes are The equations of the asymptotes 3.5.6

State the coordinates of the vertices, the coordinates of the foci, the lengths of the transverse and conjugate axes, and the equations of the asymptotes of the hyperbola defined by each equation. a) For this equation, a = 2 and b = 4. The length of the transverse axis is 2a = 4. The length of the conjugate axis is 2b = 8. The vertices are (2, 0) and (-2, 0): c 2 = a 2 + b 2 = = 20 The coordinates of the foci are The equations of the asymptotes are Analyzing an Hyperbola

b) c 2 = a 2 + b 2 = = 34 The coordinates of the foci are The equations of the asymptotes are For this equation, a = 5 and b = 3. The length of the transverse axis is 2a = 10. The length of the conjugate axis is 2b = 6. The vertices are (0, 5) and (0, -5): Analyzing an Hyperbola

The Standard Form of the Hyperbola with Centre (h, k) (h, k) The centre is (h, k). When the transverse axis is vertical, the equation in standard form is: The transverse axis is parallel to the y-axis and has a length of 2a units. The conjugate axis is parallel to the x-axis and has a length of 2b units. The slopes of the asymptotes are The general form of the equation is Ax 2 + Cy 2 + Dx + Ey + F =

When the transverse axis is horizontal, the equation in standard form is: The transverse axis is parallel to the x-axis and has a length of 2a units. The conjugate axis is parallel to the y-axis and has a length of 2b units. The slopes of the asymptotes are The Standard Form of the Hyperbola with Centre (h, k) [cont’d]

Finding the Equation of a Hyperbola The centre is (2, 3), so h = -2 and k = 3. The transverse axis is parallel to the y-axis and has a length of 10 units, so a = 5. The conjugate axis is parallel to the x-axis and has a length of 6 units, so b = 3. The vertices are (-2, 8) and (-2, -2). The slope of one asymptote is, Standard form c 2 = a 2 + b 2 = = 34 The coordinates of the foci are so a = 5 and b = 3:

Writing the Equation in General Form 9(y - 2) (x - 3) 2 = 225 9(y 2 - 4y + 4) - 25(x 2 - 6x + 9) = 225 9y y x x = x 2 +9y x - 36y = x 2 + 9y x - 36y = 0 The general form of the equation is -25x 2 + 9y x - 36y + 36 = 0 where A = -25, C = 9, D = -150, E = -36, F =

Write the equation of the hyperbola with centre at (2, -3), one vertex at (6, -3), and the coordinates of one focus at (-3, -3). The centre is (2, -3), so h = 2, k = -3. The distance from the centre to the vertex is 4 units, so a = 4. The distance from the centre to the foci is 5 units, so c = 5. Use the Pythagorean property to find b: b 2 = c 2 - a 2 = = 9 b = ± 3 9(x - 2) (y + 3) 2 = 1 9(x 2 - 4x + 4) - 16(y 2 + 6y + 9) = 144 9x x y y = 144 9x y x - 96y = 144 9x y x - 96y = 0 Standard form General form Writing the Equation of a Hyperbola

State the coordinates of the vertices, the coordinates of the foci, the lengths of the transverse and conjugate axes and the equations of the asymptotes of the hyperbola defined by 4x 2 - 9y x + 18y + 91 = 0. 4x 2 - 9y x + 18y + 91 = 0 (4x x ) + (- 9y y) + 91 = 0 4(x 2 + 8x + ____) - 9(y 2 - 2y + _____) = _____ + _____ (y - 1) 2 = -36 4(x + 4) 2 - 9(y - 1) 2 = Analyzing an Hyperbola

c 2 = a 2 + b 2 = = 13 The coordinates of the foci are The equations of the asymptotes are For this equation, a = 2 and b = 3. The length of the transverse axis is 2a = 4. The length of the conjugate axis is 2b = 6. The vertices are (-4, 3) and (-4, -1): The centre is (-4, 1) Analyzing an Hyperbola

Graph the hyperbola defined by 2x 2 - 3y 2 - 8x - 6y - 7 = 0. 2x 2 - 3y 2 - 8x - 6y - 7 = 0 (2x 2 - 8x) + (-3y 2 - 6y) - 7 = 0 2(x 2 - 4x + ____) - 3(y 2 + 2y + ___ ) = 7 + _____ + ______ (x - 2) 2 - 3(y + 1) 2 = 12 You must enter the equation in the Y= editor as y = : 2(x - 2) 2 - 3(y + 1) 2 = (y + 1) 2 = (x - 2) 2 Standard form Graphing an Hyperbola

c 2 = a 2 + b 2 = = 10 The coordinates of the foci are The equations of the asymptotes are The centre is (2, -1) Graphing the Hyperbola [cont’d]

General Effects of the Parameters A and C When A ≠ C, and A x C < 0, the resulting conic is an hyperbola. When A is positive and C is negative, the hyperbola opens to the left and right. When A is negative and C is positive, the hyperbola opens up and down. When D = E = F = 0, a degenerate occurs. E.g., 9x 2 - 4y 2 = 0 9x 2 - 4y 2 = 0 (3x - 2y)(3x + 2y) = 0 3x - 2y = 0 -2y = -3x or 3x + 2y = 0 2y = -3x These equations result in intersecting lines.

Pages A 1, 3, 6, 8, B 19, 20, 23, 25, 27, 33, 36,