AB:AE::AP:AF orDG; that is, e x m: e x : —. And A B:B E :: A P I2 n x P F-; that is, m n x : —. And Wilse y: quently,G.MorPM—PF—FG=y 71 — X r. And C G or DG—DC= In e x — s. But from the nature of the para bola G = C G X C II ; which equa tion will become that of the general for mula, by putting the literal values of those lines.
Again, if through the fixed point A you draw the indefinite right line A Q (fig. 10), parallel to P M, and you take A B = in, and draw B E = n parallel to A and through the determinate points A E, the line A E e; and if in A P you take A D = r: and draw the indefinite straight line D G parallel to A E, and take D C = s: this being done, if with the diameter C G, whose ordinates are paral lel to A P, and parameter the line C H = p, you describe a parabola C 1I ; the portion of this parabola contained in the angle B A P shall be the locus of this se cond equation, or formula : x x— 2 nyx , nnyy z X , 2n r y Ln — 7' 771 711 in -Fr r s. o.
For, if the line M Q be drawn from any point M, therein, parallel to A P ; then will AB:AE :: A Q or PM :AF or D G ; that is, 776 C and A B CL :BENAQ:QF;thatisonin:iy:Vz. And therefore G M or QM— Q F— n FG a1L e y D C = 711 And so by the common property of the parabola, you will have the foregoing se cond equation, or formula. So likewise may be found general equations for the other conic sections.
Now if it be required to draw the para bola, which we find to be the locus of this proposed equation y y —2 a y—bx -Fee=o; compare every term of the first formula with the terms of the equa tion, because y y in both is without frac.
tions ; and then will 2 because the 771 rectangle x y not being in the proposed equation, the said rectangle may be es teemed as multiplied by o; whence 71 = o, and in = because the line A E falling in A B, that is, in A P in the construc tion of the formula, the points B E do coincide. Therefore destroying all the terms adfected with in the formula, vs and substituting m for e, we shall get y y — 2 ry—px+rii+ps Again, by comparing the correspondent terms —2 r y and — 2 a y, as also — p and — b x, we have r = a, and p = b; and comparing the terms wherein are neither of the unknown quantities x y, we get r r p c c; and substituting a and cc—aa b for r and p, then will s — which is a negative expression when a is greater than c, as is here supposed. There is no need of comparing the first terms y y and y y, because they are the same. Now the values of a, r,p,s, be.
ing thus found, the sought locus may be constructed by means of the construction of the formula, and after the following manner.
Because B E = n = o (fig. 9), the points B E do coincide, and the line A E falls in A P; therefore through the fix ed point A draw the line A D = r =a parallel to P M, and draw D G parallel to AP, in which take D C — — s ; then with D C, as a diameter, whose ordinates are right lines parallel to P M, and parameter the line C = p = b, describe a parabola: then the two por tions 0 M M, It M S, contained in the angle P A 0, formed by the line A P, and the line A 0 drawn parallel to P M, will be the locus of the given equation, as is easily proved.
If in a given equation whose locus is a parabola, x x is without a fraction ; then the term of the second formula must be compared with those of the given equation.
Thus much for the method of construct ing the loci of the equations which are conic sections. If, now, an equation, whose locus is a conic section, he giv en, and the particular section whereof it is the locus be required, all the terms of the given equation being brought over to one side, so that the other be equal to nothing, there will be two cases.
Case I. When the rectangle x y is not in the given equation. 1. If either y y or vac be in the same equation, the locus will be a parabola. 2. If both x x and y y are in the equation with the same signs, the locus will be an ellipsis, or a circle. 3. If x x and y y have different signs, the locus will be an hyperbola, or the oppo site sections regarding their diameters.
Case II. When the rectangle x y is in the given equation. 1. If neither of the squares x x or y y, or only one of them, be in the same, the locus of it will be an hyperbola between the asymptotes. 2. If y y and x x be therein, having different signs, the locus will be an hyperbola re garding its diameters. 3. If both the squares x x and y y are in the equation, having the same signs, you must free the square y y from fractions; and then the locus will be an hyperbola, when the square of the fraction multiplying x y, is equal to the fraction multiplying x x ; an ellipsis, or circle, when the same is less ; and an hyperbola, or the opposite sections, regarding their diameters, when greater.