By Alan Jeffrey
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Extra resources for Mathematics for Engineers and Scientists
30) degenerates into a pair of straight line, while in the second case it reduces to a single straight line. 8 Determine the nature of the graph of the equation 7x 2 - 6yJxy + 13/-4 = 0, and sketch it. Solution Here A = 7, B = -6y'3 and C = 13, so the discriminant B 2 4AC = -256, showing that the graph is an ellipse. Now cot28= A- C = B - - 1y'3' so that 28 = rr/3, or 8 = rr/6. 20b) and substituting into the given equation reduces it to the equation of an ellipse in standard form Xz+~=l. 2 (1/2) The semi-major axis is thus 1, the semi-minor axis is 112; 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 rr/6.
Because of the necessity to use polar coordinates in problems involving differentiation, the angle 0 will always be measured in radians and, for convenience, the angle 0 will be chosen to lie in the interval 0 :5 0 < ZJC. Now let the reference line be identified with the x-axis of a cartesian coordinate system in the same plane with its origin also located at 0. Then the configuration is as shown in Fig. 18 and, if P has the cartesian coordinates (x, y) and the polar coordinates (r, 0), we see that x = r cos 0, y = r sin 0.
B>OJ X (c) Fig. 15 (a) Parabola; (b) Ellipse; (c) Hyperbola. 34 I INTRODUCTION TO SETS AND NUMBERS CH 1 The graphs of these functions are shown in Fig. 15. 22) the longest dimension 2a, which happens to lie along the x-axis, is called the major axis of the ellipse and the shortest dimension 2b which lies along the y-axis is called the minor axis. Appropriately, the quantities a and bare called, respectively, the semi-major and the semi-minor axes of the ellipse, which is symmetrical about both the major and minor axes.
Mathematics for Engineers and Scientists by Alan Jeffrey