For the more ordinary case of rectangular coordinates, w=90°, and we have simply x=p cose and y = p sin 0. Secondly, let the fixed line OB not coincide with the axis of x, but make with it an angle = a, then POB 0 and POM=0-a, Y P X M B and we have only to substitute 0-a for 0 in the preceding formulæ. For rectangular coordinates we have x=p cos (0 − a) and y = p sin (0 – a). Ex. 1. Change to polar coordinates the following equations in rectangular coordinates: x2 + y2 = 5mx. x2 - y2 = a2. Ans. Ex. 2. Change to rectangular coordinates the following equations in polar coordinates: p2 sin 20 = 2a2. p2 = a2 cos 20. p cos 10 = a*. pl = al cos 10. Ans. xy a2. 13. To express the distance between two points, in terms of their polar coordinates. Let P and Q be the two points, Q P OQ=p", QOB=0"; PQ OP2 + OQ-20P.OQ.cos POQ, = 82 = p2 + p'2 — 2p'p" cos(0′′ — 0′). B CHAPTER II. THE RIGHT LINE. 14. Any two equations between the coordinates represent geometrically one or more points. or none 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 x only; let its roots be a,, a, a, &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=α1). Let this common root be Then the values x = α1, y = B1 at once satisfy both the given equations, and denote a point which is represented by these equations. So, in like manner, is the point whose coordinates are x =α, y=ẞ1, &c. = x= Ex. 1. What point is denoted by the equations 3x+5y = 13, 4x - y = 2? Ans. x 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 a 5 + 4 = 0. The roots of this equation are a2 = 1 and 2 = 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, y=+2, y = 2, y = + 1, y=- −1. The two given equations, therefore, represent the four points 15. A single equation between the coordinates 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 coordinates of any point taken at random will not satisfy it. The assemblage then of points, whose coordinates 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 expresses that the distance of the point xy from the point (2, 3) = 4. This equation then is ati ied by the coordinates of any point on the circle whose centre is the point (2, 3), and whose radius is 4; and by the coordinates 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 coordinates signities a locus. Let us recall the construction by which (p. 1) we determined the position of a point from the two equations x = a, We took OM=a; y=b. 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 K 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 single equation xa, we should was situated somewhere on the line that line would not be determined. Hence the line MK is the locus of all the points represented by the equation x=a, were merely given the know that the point P MK, but its position in 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 coordinates, let us assume for x any value we please (x = a), 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 (x=a'), and we find, in like manner, another series of points, p', q', r', whose coordinates satisfy the equation. So again, if we assume x = a" or x = a", &c. Now, if x be supposed to take successively all 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. Ans. Giving a the values - 2, 1, 0, 1, 2, &c., we find for y, 1, 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, 2, - 4, 2. If the points thus denoted y. 2, −1, − 2, — 4, — 4, — 4, — 4, — 4, 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. Ex. 3. Represent the curve y = 3 ± √(20 − x − x2). Here to each value of x correspond two values of y. No part of the curve lies to the right of the line x = 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. * The learner is recommended to use paper ruled into little squares, which is sold under the name of logarithm paper. 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 coordinates 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 in rectangular coordinates is (Art. 4), 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 of an equation is estimated by the highest value of the sum of the indices of x and y in any term. Thus the equation xy+2x+3y=4 is of the second degree, because it contains the term xy. If this term were absent, it would be of the first degree. A curve is said to be of the nth degree when the equation which represents it is of that degree. We commence with the equation of the first degree, and we shall prove that this always represents a right line, and, conversely, that the equation of a right line is always of the first degree. 18. We have already (Art. 15) interpreted the simplest case of an equation of the first degree, namely, the equation x = a. In like manner, the equation y = b represents a line PN parallel to the axis OX, and meeting the axis OY at a distance from the origin ON=b. If we suppose b to be equal to nothing, we see that the equation y = 0 denotes the axis OX; and in like manner that x = 0 denotes the axis OY. Let us now proceed to the case next in order of simplicity, and let us examine what relation subsists between the coordinates of points situated on a right line passing through the origin. |