Also and p2 = x2+ y2, x2+y2, . F(x, y) = 0. (2) .(3) If x and y be eliminated between these three equations the required relation between p and r is obtained. This result might be at once obtained by eliminating CD from the equations and CD.p=ab, CP and CD being conjugate semi-diameters. 190. Pedal Equation deduced from Polar. Let the curve be given in Polar co-ordinates and the pole be taken at the point with regard to which it is required to find the pedal equation of the curve. Let r, be the co-ordinates of any point on the curve, and p the length of the perpendicular from the pole on the tangent at r, 0. If be the equation of the curve, then we have (see Fig. 24) Eliminate and between the equations (1), (2), (3), and the required equation between p and r will be obtained. Ex. Given m=amsin me, required its pedal equation. The following special cases of this example are worthy of notice, and will furnish exercises for the student. 191. DEF. If a perpendicular be drawn from a fixed point on a variable tangent to a curve, the locus of the M foot of the perpendicular is called the "FIRST POSITIVE PEDAL" of the original curve with regard to the given point. To find the first positive pedal with regard to the origin of any curve whose Cartesian Equation is given. Let F(x, y) = 0) be the equation of the curve. Suppose X cosa + Ysina=p touches this curve. By comparison of this equation with • (1) If x, y, λ be eliminated between the four equations (1) and (2) a result will remain which depends on p and a only. And since p, a are the polar co-ordinates of the foot of the perpendicular, if be written for p and for a, the polar equation of the locus required will be obtained. Ex. Find the first positive pedal of the curve 192. To find the Pedal with regard to the Pole of any curve whose Polar Equation is given. Let r', ' be the polar co-ordinates of the point Y, which is the foot of the perpendicular OY drawn from the pole on a tangent. Let OA be the initial line. Then Ꮎ If r, 0, be eliminated from equations 1, 2, 3, and 4 there will remain an equation in r', '. The dashes may then be dropped and the required equation will be obtained. Ex. To find the equation of the first positive pedal of the curve pmam cos mo. Taking the logarithmic differential 193. DEF. If there be a series of curves which we may designate as A, A1, A2, A3, An such that each is the first positive pedal curve of the one which immediately precedes it; then A2, Ag, etc., are respectively called the second, third, etc., positive pedals of A. Also, any one of this series of curves may be regarded as the original curve, e.g., A.; then A1⁄2 is called the first negative pedal of A„, А1 the second negative pedal, and so on. Ex. 1. Find the kth positive pedal of pmam cos mo. 1 It has been shown that the first positive pedal is 2 Ex. 2. Find the kth negative pedal of the curve pm=amcos me. |