LOCUS. 



-PF-FG = y- •-:-r,andCGorDG-DC = 

 m 



■ s. But from the nature of the parabola G M' = 



m 



C G X C H ; which equation will become that of the 

 general formuln, by putting the literal values of thofe 

 lines. 



Again : if through the fixed point A you draw the inde- 

 finite ripht line A Q {Jg. 1 8.) paralL-1 to P M, and 

 take A B = m, and draw B E r^ n, parallel to A P, and 

 through the determinate points A E, the line AE =f; 

 and if in A P you take A D = r, and draw the indefinite 

 ftraight Jine D G parallel to A E, and take D C = ^ ; this 

 being done, if with the diameter C G, whofe ordinates are 

 parallel to A P, and parameter the line C H =/i, you de- 

 fcribe a parabola C M ; this parabola ihall be the locus of 

 this fecond equation, or formula : 



- ~ J- K + — ,- y 

 m m 



, 2 n r , 



— 2 r X -{ y + r = O. 



m 



^-fy +ps. 

 m 



For if the line ?•! Q be drawn from any point M, there- 

 in, parallel to A P ; .then will A B (m) : A E (f) : : A Q 



or P M fy) : A F or D G = -^ . And A B (m) : B E 



00 : : A Q (^) : Q F ii:: -■^. And therefore G M or Q M 



-QF-FG=.v-^-r; and C G or D G - 

 m 



D C = 



12 



m 



I. And fo by the common property of 



the parabola, you will have the foregoing fecond equation, 

 or formula. So likewife may be found general equations, 

 Or formula," to the other conic fections. 



Now if it be required to draw the parabola, which we 

 find to be the locus of this propofed equation y — lay 



— b X -T c' ^^ o \ compare every term of the tirft formu- 

 la with the terms of the equation, becaufe y' in both is 



"without fractions ; and then will — = o, becaufe the reft- 



m 



iangle x y not being in the propofed equation, the faid reft- 

 angle may be efteemed as multiplied by o ; whence o = o, 

 and m = ? ; becaufe the line A E falling in A B, that is, in 

 A P in the conftruction of the formula, the points Bj E, do 

 coincide. Therefore, deftroying all the terms adfefted with 



— in the formula, and fubilitutiiig m for e, we Ihall get 



y' — zr y — p X ^ r- -ir p 1 - c. 



Again, by comparing the correfpondent terms — 2 r y, 

 and — 2 a y, as alfo — p x, and — b x, we have r ^= a, 

 aiid^ = .b ; and comparing the terms wherein are neither of 

 die unknown quantities x, y, we ^et r^ + p s := t' ; 



and fubllituting a and i for j- and p, then will s = — ~r~~-» 



which is a negative expreffion, when a is greater than c, as 

 \s here fuppofed. There is no need of comparing the firll 

 terms y- and y', becaufe they are the very fame. Now the . 

 »<Jues of CT, n, r, p, s, being ihus founds the fought locus 



may be conftruftcd by means of the coudruclion of th& for. 

 mula, and after the following manner. 



Becaufe B E («) = o (_fi^. 19.) the points B, E, do coin- 

 cide, and the hne A E falls in A P ; therefore through the 

 fixed point A draw the line AD r z= a parallel to P M, and 

 draw D G parallel to A P, in which take D C (/) = 



1 ; then with DC, as a diameter, whofe ordinates 



are right lines parallel to P M, and parameter the line C H 

 (/>) = b, defcribe a parabola : I fay, this will be the locus 

 of the given equation, as is eafily proved. If in a gi>-'cu 

 equation, whofe locus is a parabola, .v^ be without a frac- 

 tion ; then the terms of the fecond formula mull be com- 

 pared with thofe of the given equation. 



Thus much for the metiiod of condrufting the loci of 

 equations which are conic fedlions. If, now, an equation, 

 whofe locus is a conic fec^ion, be given, and the particular 

 fedion whereof it is the Iccus, be required : 



All the terms of the given equation being brought over to 

 one fide, fo that the other be equal to o, there will be two 

 cafes. 



Caji I . When the retlangle x y is not in the given equa- 

 tion. I. If either y'' or x" be in the fame equation, the 

 locus will be a parabola. 2. If both x' and y' are in the 

 equation with the fame figns, the locus will bean ellipiis, or 

 a circle. 3. If .r" and y'^ have different figns, the locus 

 will be an hyperbola, or the oppofite feftiuns regarding their 

 diameters. 



Ca/e 2. When the reftangle j; ji is in the given equation. 

 I. If neither of the fquares x' or y', or only one of them, 

 be in the fame, the locus of it will be an hyperbola between 

 the afymptotcs. 2. U y' and .v' be therein, having differ- 

 ent figns, the locus will be an hyperbola, regarding its dia- 

 meters. 3. If both the fquares x' and y' are in the equa- 

 tion, having the fame figns, then, according as the co-effi- 

 cient of x^ is greater, equal or lefs than the fquare of half 

 the co-efficient of x y, the locus fhall be an ellipfe, parabola, 

 or hyperbola. And in any cafe the locus of the equation h 

 fome conic feclion. 



We will add a problem or two, by way of illuftratian, 

 with which we mull conclude this article. 



Problem I. 



If A B be the axis of a conic feftion, from B draw B D t« 

 meet the curve in D ; and ereft D C perpendicular to A B, 

 and produce it from C till C P is in a given ratio to B D ; 

 the locus of the point P will be a conic feftion. 



I. For the ellipfe {Jig. 20.) ; put the axis A B = a, and 

 its conjugate Q O E ^ i, B C ^ .v, and tiie ratio B D 

 I C P ; : d : a. Then by the known property of the 

 ellipfe. 



CD- 



^=(--- 



confequently, B D* = 



a' 



a- - b' 



^ — ■ jr', and, therefore, 



a 



b^ax + {a^ - b^) x' a^ - 



+ x' = 



+ 



C P^ 



E D' = 



• ( '~_ i: " ~ *' )> wliich. 



I 



if a be great-r than b, is an equation to tlie hyperbola, the 

 and 



axes of which are -„— 



b' "■'" d ^ (a' - 6')' 

 And if b be greater than a, the equation beciames 

 I 1 2 



-cr 



