Prime Coe cients Formula for Rotating Conics

Transcription

1 Prime Coe cients Formula for Rotating Conics David Rose Polk State College David Rose (Institute) Prime Coe cients Formula for Rotating Conics 1 / 18

2 It is known that the graph of a conic section in the x, y -coordinate plane has a second degree equation in the variables x and y. I.e., Ax + Bxy + Cy + Dx + Ey + F = 0. It is also known that rotation of the axes through an angle about the origin, transforms the equation to A 0 x + B 0 xy + C 0 y + D 0 x + E 0 y + F 0 = 0. And, if B 6= 0, then rotation of the axes through an angle θ = 1 A C cot 1 yields B 0 = 0 so that B in this rotated system the conic has vertical and horizontal axes of symmetry unless it is a parabola in which case the axis of symmetry is either vertical or horizontal. The rotation of axes coordinate transformations are as follows. x = x 0 cos θ y 0 sin θ y = x 0 sin θ + y 0 cos θ David Rose (Institute) Prime Coe cients Formula for Rotating Conics / 18

3 By substitution into Ax + Bxy + Cy + Dx + Ey + F = 0 we obtain A 0 x + B 0 xy + C 0 y + D 0 x + E 0 y + F 0 = 0 with the following prime coe cients. A 0 = A cos θ + B sin(θ) + C sin θ B 0 = B cos(θ) (A C ) sin(θ) C 0 = A sin B θ sin(θ) + C cos θ D 0 = D cos θ + E sin θ E 0 = E cos θ D sin θ F 0 = F David Rose (Institute) Prime Coe cients Formula for Rotating Conics 3 / 18

4 Of course, B 0 = 0 if θ = 1 A C cot 1. Then 0 < θ < π and also, B sin θ > 0 and cos θ > 0 since 0 < θ < π. From the gure, whether θ is acute or obtuse, s A cos(θ) = A C BH and sin(θ) = 1 C H where H = + 1. B David Rose (Institute) Prime Coe cients Formula for Rotating Conics 4 / 18

5 Therefore, cos θ = 1 (A (1 + cos(θ)) = C sin θ = 1 BH (A C ) (1 cos(θ)) = BH positive, we have ) + BH BH and. Since sin θ and cos θ are Now, sin θ = r BH (A C ) BH and cos θ = r (A C ) + BH. BH A 0 = A cos θ + B sin(θ) + C sin θ (A C ) + BH = A + B BH (A C ) BH BH + C BH = (A C ) + B + (A + C )BH BH = B H + (A + C )BH BH = A + C + BH. David Rose (Institute) Prime Coe cients Formula for Rotating Conics 5 / 18

6 Similarly, C 0 = A + C BH and the full formula is as follows. A 0 = A + C + BH, B 0 = 0, C 0 = A + C BH r r BH + (A C ) BH (A C ) D 0 = D + E, r BH r BH BH + (A C ) BH (A C ) E 0 = E D, BH BH F 0 = F David Rose (Institute) Prime Coe cients Formula for Rotating Conics 6 / 18

7 It follows that F and A + C are invariant under rotation of axes by our angle θ. For A 0 + C 0 = A + C + BH + A + C BH = A + C. Also, the discriminant B 4AC is invariant. B 0 4A 0 C 0 = A + C + BH A + C BH 4 = [BH (A + C )][BH + (A + C )] = B H (A + C ) = B + (A C ) (A + C ) = B 4AC If the graph is a hyperbola, then 4A 0 C 0 = B 4AC > 0 and if the graph is an ellipse 4A 0 C 0 = B 4AC < 0. And, if the graph is a parabola, then 4A 0 C 0 = B 4AC = 0. David Rose (Institute) Prime Coe cients Formula for Rotating Conics 7 / 18

8 Example Identify p as a conic section and sketch the graph of x 3xy + y x = 0. We have A = 1, B = p 3, C =, D =, and E = F = 0. For θ = 1 A C cot 1 = 1 1p3 B cot 1 = 30 o we have BH = p s 1 3 p + 1 = and B 4AC = 5 < 0 so the 3 graph cannot be either a parabola or a hyperbola, but may be an ellipse. Invoking the prime coe cients formula we have A 0 = 1, B0 = 0, C 0 = 5, D0 = p 3, E 0 = 1, and F 0 = 0. The equation relative to the rotated coordinate system is 1 x y 0 p 3x 0 + y 0 = 0. David Rose (Institute) Prime Coe cients Formula for Rotating Conics 8 / 18

9 or equivalently, x 0 + 5y 0 p 3x 0 + y 0 = 0. p Completing the square we have x y = , or x 0 p 3 4 p5 + y = 1. 5 David Rose (Institute) Prime Coe cients Formula for Rotating Conics 9 / 18

10 Indeed, the graph is a horizontal ellipse in the rotated system centered at p3, (x 0, y 0 1 ) =. In the original coordinate system, the center is 5 (x, y) = ( p 3 cos 30 o sin 30o, p 3 sin 30 o 1 5 cos 30o ) 3 = + 1 p p! , = , p! 3. 5 In this coordinate system the ellipse is not horizontal but cocked or tilted 30 o counterclockwise. The x-intercepts are (0, 0) and (, 0). The only y-intercept is (0, 0). The ellipse is tangent to the y-axis at the origin. David Rose (Institute) Prime Coe cients Formula for Rotating Conics 10 / 18

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12 Example Recognize the graph of y = x as a hyperbola. We have equivalently, x for x 6=, xy x y = 0 so that A = C = 0, B = 1, D = 1, E =, and F = 0. If θ = 1 cot 1 (0) = 45 o is the angle of rotation of the x, y-coordinate system, we have BH = 1 so that A 0 = 1, B0 = 0, C 0 = 1, D0 = 3 p, E 0 = 1 p, and F 0 = 0. This yields the rotated equation 1 x 0 1 y 0 3 p x 0 1 p y 0 = 0 or x 0 y 0 3 p x 0 p y 0 = 0. David Rose (Institute) Prime Coe cients Formula for Rotating Conics 1 / 18

13 Completing the square gives us x 0 3 p y p = 9 1 = 4 so that indeed the graph is a hyperbola with rotated equation x 0 3 p y p = 1. David Rose (Institute) Prime Coe cients Formula for Rotating Conics 13 / 18

14 3 The center is at (x 0, y 0 ) = p, 1 p in the rotated system which in the unrotated system is 3p (x, y) = cos 45o + p 1 sin 45 o 3, p sin 45 o 1 p cos 45 o 3 = + 1, 3 1 = (, 1). The asymptotes in the rotated system are the lines y 0 + p 1 = x 0 3 p which leads to y 0 = x 0 p or y 0 = x 0 + p. David Rose (Institute) Prime Coe cients Formula for Rotating Conics 14 / 18

15 Now, by rotating the x 0, y 0 -system through the angle θ we have x 0 = x cos( θ) y sin( θ) and y 0 = x sin( θ) + y cos( θ) or x 0 = x cos θ + y sin θ, and y 0 = x sin θ + y cos θ. Thus, the equations of the asymptotes in the x, y-system are x sin 45 o + y cos 45 o = x cos 45 o + y sin 45 o p or x sin 45 o + y cos 45 o = x cos 45 o y sin 45 o + p. David Rose (Institute) Prime Coe cients Formula for Rotating Conics 15 / 18

16 These equations simplify to p ( x + y) = p (x + y 4) or or x = or y = 1 p p ( x + y) = ( x y + ) which agrees with the known asymptotes for this rational function. Similarly, we can nd the two lines of symmetry for the graph of our rational function in the x, y-system. David Rose (Institute) Prime Coe cients Formula for Rotating Conics 16 / 18

17 In the x 0, y 0 -system, the lines of symmetry are x 0 = 3 p and y 0 = In the x, y-system these are the equations 1 p. x cos 45 o + y sin 45 o = 3 p and x sin 45 o + y cos 45 o = 1 p which simplify to x + y = 3 and x y = 1. David Rose (Institute) Prime Coe cients Formula for Rotating Conics 17 / 18

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