Rotation of Axes 2 ROTATION OF AXES

Rotation of Axes

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 Cy2 Dx Ey F 0

Here we show that the general second-degree equation

1 Ax2 Bxy Cy2 Dx Ey 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, y in the first coordinate system and X, Y 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 cos y r sin

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 cos rcos cos sin sin

r cos cos r sin sin cos Y sin

A similar computation gives y in terms of X and Y and so we have the following formulas:

2 x X cos Y sin y X sin Y cos

By solving Equations 2 for X and Y we obtain

3 X x cos y sin Y x sin y cos ROTATION OF AXES 3

EXAMPLE 1 If the axes are rotated through 60 , find the XY -coordinates of the point whose xy-coordinates are 2, 6 .

SOLUTION Using Equations 3 with x 2 ,y 6 , and 60 , we have X 2 cos 606 sin 60 1 3s3

Y 2 sin 606 cos 60 s3 3

The XY-coordinates are (1 3s3, 3 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

AX cos Y sin 2 BX cos Y sin sin Y cos

CX sin Y cos 2 DX cos Y sin

EX sin Y cos F 0

Expanding and collecting terms, we obtain an equation of the form

4 AX 2 BXY CY 2 DX EY F 0 where the coefficient B of XY is

B 2C A sin cos Bcos2 sin2

C A sin 2 B cos 2

To eliminate the XY term we choose so that B 0 , that is,

A C sin 2 cos 2 or

A 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 C cot 2 0 B 4 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 y s2 s2 s2 s2 π 4 Substituting these expressions into the original equation gives 0 x X Y X Y X 2 Y 2 1 or 1 s2 s2 s2 s2 2 2

We recognize this as a hyperbola with vertices (s2, 0) in the XY -coordinate sys- FIGURE 3 tem. The asymptotes are Y X 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 72xy 52y 2 30x 40y 75 0

SOLUTION This equation is in the form of Equation 1 with A 73 ,B 72 , and C 52. Thus A C 73 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 cos 2 1 7 4 cos 25 2 2 5

1 cos 2 1 7 3 sin 25 2 2 5

The rotation equations (2) become 4 3 3 4 x 5 X 5 Y y 5 X 5 Y Substituting into the given equation, we have 4 3 2 4 3 3 4 3 4 2 73(5 X 5Y) 72(5 X 5Y)(5 X 5Y) 52(5 X 5Y)

4 3 3 4 30(5 X 5Y) 40(5 X 5Y) 75 0 which simplifies to 4X 2 Y 2 2Y 3 Completing the square gives

Y 12 4X 2 Y 12 4 or X 2 1 4

and we recognize this as being an ellipse whose center is 0, 1 in XY -coordinates. ROTATION OF AXES 5

1 4 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

A Click here for answers. (c) Find an equation of the directrix in the xy -coordinate system. 1–4 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, 4, 30 2. 4, 3, 45 2x 2 72xy 23y 2 80x 60y 125 3.2, 4, 60 4. 1, 1, 15 represents a hyperbola.

(b) Find the XY -coordinates of the foci. Then find the xy-coordinates of the foci. 5–12 Use rotation of axes to identify and sketch the curve. (c) Find the xy -coordinates of the vertices. 5. x 2 2xy y 2 x y 0 (d) Find the equations of the asymptotes in the xy-coordinate system. 2 2 6. x xy y 1 (e) Find the eccentricity of the hyperbola. 2 2 7. x xy y 1 15. Suppose that a rotation changes Equation 1 into Equation 4. 8. s3xy y 2 1 Show that 9. 97x 2 192xy 153y 2 225 A C A C 2 s 2 10. 3x 12 5xy 6y 9 0 16. Suppose that a rotation changes Equation 1 into Equation 4. 11. 2s3xy 2y 2 s3x y 0 Show that 2 2 12. 16x 2 8s2xy 2y 2 (8s2 3)x (6s2 4)y 7 B 4A C B 4AC

17. Use Exercise 16 to show that Equation 1 represents (a) a 2 2 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 4AC 0, except in degenerate 36x 2 96xy 64y 2 20x 15y 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 9Ð12. 6 ANSWERS

Answers

2 2 1. ((s3 4)2, (4s3 1)2) 9. X Y 9 1, ellipse y 3. (2s3 1, s3 2) X 5. X s2Y 2, parabola Y

4 sin–!” ’ y 5 0 x Y X

1 1 x 0 11. X 12 3Y 2 1, hyperbola y Y 7. 3X 2 Y 2 2, ellipse X

y 0 x Y X

0 x

2 17 17 51 13. (a)Y 1 4X (b) (0, 16 ), ( 20 , 80 ) (c) 64x 48y 75 0