Rotation of Axes 2 I ROTATION OF AXES Rotation of Axes GGGGGGGGGGGGGGGG L For a discussion of conic sections, see In precalculus or calculus you may have studied conic sections with equations of the Calculus, Fourth Edition, Section 11.6 form Calculus, Early Transcendentals, Fourth Edition, Section 10.6 Ax2 ϩ Cy2 ϩ Dx ϩ Ey ϩ F ෇ 0 Here we show that the general second-degree equation 1 Ax2 ϩ Bxy ϩ Cy2 ϩ Dx ϩ Ey ϩ F ෇ 0 can be analyzed by rotating the axes so as to eliminate the term Bxy. In Figure 1 the x and y axes have been rotated about the origin through an acute angle ␪ to produce the X and Y axes. Thus, a given point P has coordinates ͑x, y͒ in the first coordinate system and ͑X, Y͒ in the new coordinate system. To see how X and Y are related to x and y we observe from Figure 2 that X ෇ r cos ␾ Y ෇ r sin ␾ x ෇ r cos͑␪ ϩ ␾͒ y ෇ r sin͑␪ ϩ ␾͒ y y Y P(x, y) Y P(X, Y) P X Y X r y ˙ ¨ ¨ 0 x 0 x X x FIGURE 1 FIGURE 2 The addition formula for the cosine function then gives x ෇ r cos͑␪ ϩ ␾͒ ෇ r͑cos ␪ cos ␾ Ϫ sin ␪ sin ␾͒ ෇ ͑r cos ␾͒ cos ␪ Ϫ ͑r sin ␾͒ sin ␪ ෇ ⌾ cos ␪ Ϫ Y sin ␪ A similar computation gives y in terms of X and Y and so we have the following formulas: 2 x ෇ X cos ␪ Ϫ Y sin ␪ y ෇ X sin ␪ ϩ Y cos ␪ By solving Equations 2 for X and Y we obtain 3 X ෇ x cos ␪ ϩ y sin ␪ Y ෇ Ϫx sin ␪ ϩ y cos ␪ ROTATION OF AXES N 3 EXAMPLE 1 If the axes are rotated through 60Њ, find the XY-coordinates of the point whose xy-coordinates are ͑2, 6͒. SOLUTION Using Equations 3 with x ෇ 2,y ෇ 6 , and ␪ ෇ 60Њ, we have X ෇ 2 cos 60Њϩ6 sin 60Њ ෇ 1 ϩ 3s3 Y ෇ Ϫ2 sin 60Њϩ6 cos 60Њ ෇ Ϫs3 ϩ 3 The XY-coordinates are (1 ϩ 3s3, 3 Ϫ s3). Now let’s try to determine an angle ␪ such that the term Bxy in Equation 1 disap- pears when the axes are rotated through the angle ␪. If we substitute from Equations 2 in Equation 1, we get A͑X cos ␪ Ϫ Y sin ␪͒2 ϩ B͑X cos ␪ Ϫ Y sin ␪͒͑⌾ sin ␪ ϩ Y cos ␪͒ ϩ C͑X sin ␪ ϩ Y cos ␪͒2 ϩ D͑X cos ␪ Ϫ Y sin ␪͒ ϩ E͑X sin ␪ ϩ Y cos ␪͒ ϩ F ෇ 0 Expanding and collecting terms, we obtain an equation of the form 4 AЈX 2 ϩ BЈXY ϩ CЈY 2 ϩ DЈX ϩ EЈY ϩ F ෇ 0 where the coefficient BЈ of XY is BЈ ෇ 2͑C Ϫ A͒ sin ␪ cos ␪ ϩ B͑cos2␪ Ϫ sin2␪͒ ෇ ͑C Ϫ A͒ sin 2␪ ϩ B cos 2␪ To eliminate the XY term we choose ␪ so that BЈ ෇ 0, that is, ͑A Ϫ C͒ sin 2␪ ෇ 〉 cos 2␪ or A Ϫ C 5 cot 2␪ ෇ B EXAMPLE 2 Show that the graph of the equation xy ෇ 1 is a hyperbola. SOLUTION Notice that the equation xy ෇ 1 is in the form of Equation 1 where A ෇ 0, B ෇ 1, and C ෇ 0. According to Equation 5, the xy term will be eliminated if we choose ␪ so that A Ϫ C cot 2␪ ෇ ෇ 0 B 4 I ROTATION OF AXES X@ Y@ ␪ ෇ ␲͞ ␪ ෇ ␲͞ ␪ ෇ ␪ ෇ ͞ xy=1 or - =1 This will be true if 2 2, that is,4 . Then cos sin 1 s2 and 2 2 y Equations 2 become X Y X Y X Y x ෇ Ϫ y ෇ ϩ s2 s2 s2 s2 π 4 Substituting these expressions into the original equation gives 0 x X Y X Y X 2 Y 2 ͩ Ϫ ͪͩ ϩ ͪ ෇ 1 or Ϫ ෇ 1 s2 s2 s2 s2 2 2 We recognize this as a hyperbola with vertices (Ϯs2, 0) in the XY-coordinate sys- FIGURE 3 tem. The asymptotes are Y ෇ ϮX in the XY-system, which correspond to the coor- dinate axes in the xy-system (see Figure 3). EXAMPLE 3 Identify and sketch the curve 73x 2 ϩ 72xy ϩ 52y 2 ϩ 30x Ϫ 40y Ϫ 75 ෇ 0 SOLUTION This equation is in the form of Equation 1 with A ෇ 73,B ෇ 72 , and C ෇ 52. Thus A Ϫ C 73 Ϫ 52 7 cot 2␪ ෇ ෇ ෇ 25 24 B 72 24 From the triangle in Figure 4 we see that 2¨ ␪ ෇ 7 7 cos 2 25 FIGURE 4 The values of cos ␪ and sin ␪ can then be computed from the half-angle formulas: 1 ϩ cos 2␪ 1 ϩ 7 4 cos ␪ ෇ ͱ ෇ ͱ 25 ෇ 2 2 5 1 Ϫ cos 2␪ 1 Ϫ 7 3 sin ␪ ෇ ͱ ෇ ͱ 25 ෇ 2 2 5 The rotation equations (2) become ෇ 4 Ϫ 3 ෇ 3 ϩ 4 x 5 X 5 Y y 5 X 5 Y Substituting into the given equation, we have 4 Ϫ 3 2 ϩ 4 Ϫ 3 3 ϩ 4 ϩ 3 ϩ 4 2 73(5 X 5Y) 72(5 X 5Y)(5 X 5Y) 52(5 X 5Y) ϩ 4 Ϫ 3 Ϫ 3 ϩ 4 Ϫ ෇ 30(5 X 5Y) 40(5 X 5Y) 75 0 which simplifies to 4X 2 ϩ Y 2 Ϫ 2Y ෇ 3 Completing the square gives ͑Y Ϫ 1͒2 4X 2 ϩ ͑Y Ϫ 1͒2 ෇ 4 or X 2 ϩ ෇ 1 4 and we recognize this as being an ellipse whose center is ͑0, 1͒ in XY-coordinates. ROTATION OF AXES N 5 ␪ ෇ Ϫ1 4 Ϸ Њ Since cos (5) 37 , we can sketch the graph in Figure 5. y X Y (0, 1) ¨Å37° 0 x 73≈+72xy+52¥+30x-40y-75=0 or FIGURE 5 4X@+(Y-1)@=4 Exercises GGGGGGGGGGGGGGGGGGGGGGGGGG A Click here for answers. (c) Find an equation of the directrix in the xy-coordinate system. 1–4 I Find the XY-coordinates of the given point if the axes (a) Use rotation of axes to show that the equation are rotated through the specified angle. 14. 1.͑1, 4͒, 30Њ 2. ͑4, 3͒, 45Њ 2x 2 Ϫ 72xy ϩ 23y 2 Ϫ 80x Ϫ 60y ෇ 125 3.͑Ϫ2, 4͒, 60Њ 4. ͑1, 1͒, 15Њ represents a hyperbola. IIIIIIIIIIIIIIIIIIII (b) Find the XY-coordinates of the foci. Then find the xy-coordinates of the foci. 5–12 I Use rotation of axes to identify and sketch the curve. (c) Find the xy-coordinates of the vertices. 5. x 2 Ϫ 2xy ϩ y 2 Ϫ x Ϫ y ෇ 0 (d) Find the equations of the asymptotes in the xy-coordinate system. 2 2 6. x Ϫ xy ϩ y ෇ 1 (e) Find the eccentricity of the hyperbola. 2 ϩ ϩ 2 ෇ 7. x xy y 1 15. Suppose that a rotation changes Equation 1 into Equation 4. 8. s3xy ϩ y 2 ෇ 1 Show that Јϩ Ј ෇ ϩ 9. 97x 2 ϩ 192xy ϩ 153y 2 ෇ 225 A C A C 2 Ϫ s ϩ 2 ϩ ෇ 10. 3x 12 5xy 6y 9 0 16. Suppose that a rotation changes Equation 1 into Equation 4. 11. 2s3xy Ϫ 2y 2 Ϫ s3x Ϫ y ෇ 0 Show that ͑ Ј͒2 Ϫ Ј Ј ෇ 2 Ϫ 12. 16x 2 Ϫ 8s2xy ϩ 2y 2 ϩ (8s2 Ϫ 3)x Ϫ (6s2 ϩ 4)y ෇ 7 B 4A C B 4AC IIIIIIIIIIIIIIIIIIII 17. Use Exercise 16 to show that Equation 1 represents (a) a 2 Ϫ ෇ 2 Ϫ Ͻ 13. (a) Use rotation of axes to show that the equation parabola if B 4AC 0, (b) an ellipse if B 4AC 0, and (c) a hyperbola if B 2 Ϫ 4AC Ͼ 0, except in degenerate 36x 2 ϩ 96xy ϩ 64y 2 ϩ 20x Ϫ 15y ϩ 25 ෇ 0 cases when it reduces to a point, a line, a pair of lines, or no graph at all. represents a parabola. (b) Find the XY-coordinates of the focus. Then find the 18. Use Exercise 17 to determine the type of curve in xy-coordinates of the focus. Exercises 9–12. 6 I ANSWERS Answers GGGGGGGGGGGGGGGGGGGGGGGGGG 2 2 1. ((s3 ϩ 4)͞2, (4s3 Ϫ 1)͞2) 9. X ϩ ͑Y ͞9͒ ෇ 1, ellipse y 3. (2s3 Ϫ 1, s3 ϩ 2) X 5. X ෇ s2Y 2, parabola Y 4 sin–!” ’ y 5 0 x Y X 1 1 x 0 11. ͑X Ϫ 1͒2 Ϫ 3Y 2 ෇ 1, hyperbola y Y 7. 3X 2 ϩ Y 2 ෇ 2, ellipse X y 0 x Y X 0 x Ϫ ෇ 2 17 Ϫ17 51 13. (a)Y 1 4X (b) (0, 16 ), ( 20 , 80 ) (c) 64x Ϫ 48y ϩ 75 ෇ 0.