Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

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1 Rotation of Aes

2 ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the general second-degree equation 1 A B C D E F can be analzed b rotating the aes so as to eliminate the term B. In Figure 1 the and aes have been rotated about the origin through an acute angle to produce the and aes. Thus, a given point P has coordinates, in the first coordinate sstem and, in the new coordinate sstem. To see how and are related to and we observe from Figure that r cos r cos r sin r sin P(, ) P(, ) r P FIGURE 1 FIGURE The addition formula for the cosine function then gives r cos rcos cos sin sin r cos cos r sin sin cos sin A similar computation gives in terms of and and so we have the following formulas: cos sin sin cos B solving Equations for and we obtain 3 cos sin sin cos

3 ROTATION OF AES 3 EAMPLE 1 If the aes are rotated through 6, find the -coordinates of the point whose -coordinates are, 6. 6 SOLUTION Using Equations 3 with, 6, and, we have cos 6 6 sin 6 1 3s3 sin 6 6 cos 6 s3 3 The -coordinates are (1 3s3, 3 s3). Now let s tr to determine an angle such that the term B in Equation 1 disappears when the aes are rotated through the angle. If we substitute from Equations in Equation 1, we get A cos sin B cos sin sin cos C sin cos D cos sin E sin cos F Epanding and collecting terms, we obtain an equation of the form A B C D E F where the coefficient B of is B C A sin cos Bcos sin C A sin B cos To eliminate the term we choose so that B, that is, A C sin cos or 5 cot A C B EAMPLE Show that the graph of the equation 1 is a hperbola. SOLUTION Notice that the equation 1 is in the form of Equation 1 where A, B 1, and C. According to Equation 5, the term will be eliminated if we choose so that cot A C B

4 ROTATION =1 or - =1 This will be true if, that is,. Then cos sin 1s and Equations become s s s s π Substituting these epressions into the original equation gives s s 1 or s s 1 FIGURE 3 We recognize this as a hperbola with vertices (s, ) in the -coordinate sstem. The asmptotes are in the -sstem, which correspond to the coordinate aes in the -sstem (see Figure 3). EAMPLE 3 Identif and sketch the curve SOLUTION This equation is in the form of Equation 1 with A 73, B 7, and C 5. Thus 5 cot A C B From the triangle in Figure we see that cos 7 5 FIGURE The values of cos and sin can then be computed from the half-angle formulas: cos sin 1 cos 1 cos The rotation equations () become Substituting into the given equation, we have 73( 5 3 5) 7( 5 3 5)( 3 5 5) 5( 3 5 5) 3( 5 3 5) ( 3 5 5) 75 which simplifies to Completing the square gives 3 1 or 1 1 and we recognize this as being an ellipse whose center is, 1 in -coordinates.

5 ROTATION OF AES 5 cos 1 ( 5) 37 Since, we can sketch the graph in Figure 5. FIGURE 5 (, 1) Å = Eercises A Click here for answers. 1 Find the -coordinates of the given point if the aes are rotated through the specified angle. 1. 1,, 3., 3, 5 3.,, 6. 1, 1, Use rotation of aes to identif and sketch the curve s s s3 s s (8s 3) (6s ) (a) Use rotation of aes to show that the equation represents a parabola. (b) Find the -coordinates of the focus. Then find the -coordinates of the focus. (c) Find an equation of the directri in the -coordinate sstem. 1. (a) Use rotation of aes to show that the equation represents a hperbola. (b) Find the -coordinates of the foci. Then find the -coordinates of the foci. (c) Find the -coordinates of the vertices. (d) Find the equations of the asmptotes in the -coordinate sstem. (e) Find the eccentricit of the hperbola. 15. Suppose that a rotation changes Equation 1 into Equation. Show that A C A C 16. Suppose that a rotation changes Equation 1 into Equation. Show that B AC B AC 17. Use Eercise 16 to show that Equation 1 represents (a) a parabola if B AC, (b) an ellipse if B AC, and (c) a hperbola if B AC, ecept in degenerate cases when it reduces to a point, a line, a pair of lines, or no graph at all. 18. Use Eercise 17 to determine the tpe of curve in Eercises 9 1.

6 6 ANSWERS Answers 1. ((s3 ), (s3 1)) , ellipse (s3 1, s3 ) s, parabola sin! , hperbola 7. 3, ellipse 13. (a) 1 (b) (c) (, 17 16), ( 17, 51 8)