By Alan Jeffrey
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Extra resources for Mathematics for Engineers and Scientists, Sixth Edition
44b) and substituting into the given equation reduces it to the equation of an ellipse in standard form / / / Y2 X 2 ' 2 0 1: 1 2 The semi-major axis is thus 1, the semi-minor axis is 12; the centre is at the origin and the X- and Y-axes are obtained from the x- and y-axes by means of an anticlockwise rotation about the origin through an angle p/6. The graph is shown in Fig. 18. j pﬃﬃﬃ Fig. 9 Cone Let G be a simple closed plane curve and V some fixed point not in the plane of G. Then a cone is the three-dimensional surface generated by a straight line passing through V which is allowed to move so that its point of intersection P with G traverses G.
Inspection of Fig. 14(b) shows that XP 0 xP ( a and YP 0 yp ( b, so that dropping the suffix P we arrive at the general result / X 0 x(a; / / / (1:43) Y 0 y(b: This is the required coordinate transformation. If, for example, the axes in Fig. 39) would become X2'Y2 0 r2. 5 Rotation of axes In certain circumstances it is necessary to rotate Cartesian axes about the origin through an angle u as shown in Fig. 16. If the new coordinate axes are O(X, Y) it is necessary to know the relationship that exists between the coordinates of the same representative point P in each Fig.
Some useful results obtained by applying the binomial expansion when n is not an integer are as follows: 1 1:3 2 1:3:7 3 (19x)1=4 0 19 x( x9 x (. . ; 4 4:8 4:8:12 for x2 5 1; 1 1:1 2 1:1:3 3 x9 x (. . ; (19x)1=2 0 19 x( 2 2:4 2:4:6 for x2 5 1; 1 1:3 2 1:3:5 3 x x '. . ; (19x)(1=2 0 1 x' 2 2:4 2:4:6 for x2 B 1; (19x)(1 0 1x'x2 x3 '. . ; for x2 B 1: 24 Numbers, trigonometric functions and coordinate geometry In these expansions the 9 sign has been used to combine into a single expression the separate cases corresponding to the ' sign and the ( sign.