If we were given a fixed point 0, and a fixed line through it P OB, it is evident that we should know the position of any point P, if we knew the length OP, and also the angle POB. The line OP is called the radius vector; the fixed point is called the pole; and this method is called the method of polar coordinates. B It is very easy, being given the x and y co-ordinates of a point, to find its polar ones, Y or vice versâ. For the more ordinary case of rectangular co-ordinates, w= 90°, and we have simply and we have only to substitute a for 0 in the preceding formulæ. For rectangular co-ordinates we have x = p cos (0 - a) and y = p sin (0 − a). Ex. 1. Change to polar co-ordinates the following equations in rectangular co-ordinates: x2 + y2 = 5mx. x2 - y2 = a2. Ex. 2. Change to rectangular co-ordinates the following equations in polar co-ordinates: 13. To express the distance between two points, in terms of 14. Any two equations between the co-ordinates represent geometrically one or more points. If the equations be both of the first degree (see Ex. 5, p. 4), they denote a single point. For solving the equations for x and y, we obtain a result of the form xa, y=b, which, as was proved in the last chapter, represents a point. If the equations be of higher degree, they represent more points than one. For, eliminating y between the equations, we obtain an equation containing a only; let its roots be a,,. aga, &c. Now, if we substitute any of these values (a,) for x in the original equations, we get two equations in y, which must have a common root (since the result of elimination between the equations is rendered 0 by the supposition x = a,). Let this common root be y = B. Then the values x = a1, y=ß1, at once satisfy both the given equations, and denote a point B which is represented by these equations. So, in like manner, is the point whose co-ordinates are x = α, y = ẞ1, &c. Ex. 1. What point is denoted by the equations 3x+5y= 13, 4x - y = 2? Ans. 1, y = 2. = Ex. 2. What points are represented by the two equations x2 + y2 = 5, xy = 2? Eliminating y between the equations, we get x1 - 5x2 + 4 = 0. The roots of this equation are x = 1 and r2 = 4, and, therefore, the four values of x are x = + 1, x = 1, x = +2, x = − 2. Substituting these successively in the second equation, we obtain the corresponding values of y, The two given equations, therefore, represent the four points (+1, + 2), (1, 2), (+2, + 1), (-2, -1). Ex. 3. What points are denoted by the equations x - y = 1, x2 + y2 = 25 ? Ans. (4, 3), (- 3, - 4). Ex. 4. What points are denoted by the equations Ans. (1, 1), (2, 3), (3, 3), (4, 1). 15. A single equation between the co-ordinates denotes a geometrical locus. One equation evidently does not afford us conditions enough to determine the two unknown quantities x, y; and an indefinite number of systems of values of x and y can be found which will satisfy the given equation. And yet the co-ordinates of any point taken at random will not satisfy it. The assemblage then of points, whose co-ordinates do satisfy the equation, forms a locus, which is considered the geometrical signification of the given equation. Thus, for example, we saw (Ex. 3, p. 4), that the equation (x − 2)2 + (y − 3)2 = 16 expresses that the distance of the point ay from the point (2, 3) = 4. This equation then is satisfied by the co-ordinates of any point on the circle whose centre is the point (2, 3), and whose radius is 4; and by the co-ordinates of no other point. This circle then is the locus which the equation is said to represent. We can illustrate by a still simpler example, that a single equation between the co-ordinates signifies a locus. Let us recall the construction by which (p. 1) we determined position of a point from y=b. We took OM= a; we drew MK parallel to OY; and then, measuring MP=b, we found P, the point required. Had we been given a different value of y, x=a, y=b', we should proceed as before, and we Y the K P' P should find a point P' still situated on the line MK, but at a different distance from M. Lastly, if the value of y were left wholly indeterminate, and we were merely given the single equation x = a, we should know that the point P was situated somewhere on the line MK, but its position in Hence the line MK is that line would not be determined. the locus of all the points represented by the equation x=a, since, whatever point we take on the line MK, the x of that point will always = a. 16. In general, if we are given an equation of any degree between the co-ordinates, let us assume for x any value we please (xa), and the equation will enable us to determine a finite number of values of y answering to this particular value of x; and consequently, the equation will be satisfied for each of the points (p, q, r, &c.), whose x is the assumed value, and whose y is that found from the equation. Again, assume for x any other value (a), and we find, ano in like manner, ther series of points, P', q', r', whose coordinates satisfy the equation. So again, if we assume x=a' a" orxa""&c. Now, if x be supposed to take successively all PPP a a' a" a possible values, the assemblage of points found as above will form a locus, every point of which satisfies the conditions of the equation, and which is, therefore, its geometrical signification. We can find in the manner just explained as many points of this locus as we please, until we have enough to represent its figure to the eye. Ex. 1. Represent in a figure a series of points which satisfy the equation y = 2x + 3. 1, 3, 5, 7, &c., and the corresponding points will be seen all to lie on a right line. Ex. 2. Represent the locus denoted by the equation y = x2 - 3x - 2. Ans. To the values for x, −1, − 1, 0, 1, 1, 1, 2, §, 3, 1, 4; correspond for y, 2, −1, −2, 19, -4, -17, 4, 13, -2, 1, 2. If the points thus denoted be laid down on paper, they will sufficiently exhibit the form of the curve, which may be continued indefinitely by giving a greater positive or negative values. x2). Ex. 3. Represent the curve y = 3 ± √(20− x Here to each value of a correspond two values of y. No part of the curve lies to the right of the line = 4, or to the left of the line x = - 5, since by giving greater positive or negative values to x, the value of y becomes imaginary. 17. The whole science of Analytic Geometry is founded on the connexion which has been thus proved to exist between an equation and a locus. If a curve be defined by any geometrical property, it will be our business to deduce from that property an equation which must be satisfied by the co-ordinates of every point on the curve. Thus, if a circle be defined as the locus of a point (x, y), whose distance from a fixed point (a, b) is constant, and equal to r; then the equation of the circle to rectangular co-ordinates, is (Art. 4), (x − a)* + (y — b)2 = r2. On the other hand, it will be our business when an equation is given, to find the figure of the curve represented, and to deduce its geometrical properties. In order to do this systematically, we make a classification of equations according to their degrees, and beginning with the simplest, examine the form and properties of the locus represented by the equation. The degree The learner is recommended to use paper ruled into little squares, which is sold under the name of logarithm paper. |