PLANE ANALYTICA L.] distance Ir, or along the line considered as drawn reversely from .2 towards z’, the equal negative distance — lit, and the expression for the distance p is thus properly of the form
- 1: It. It is interesting to compare expressions which do
not involve a radical: thus, in seeking for the expression for the perpendicular distance of the point (a, b) from a given line, let the equation of the given line be taken in the form, ac cos a+ 3/ sin a —p=0 (p being the perpen- dicular distance from the origin, a its inclination to the axis of .r): the equation of the line may also be written (.c — u) cos a+ (3/— b) sin a—p,=0, and we have thence )2, =1» — w cos a— l) sin a, the required expression for the distance 7),: it is here assumed that p, is drawn from (It, 1;) in the same sense as p is drawn from the origin, and the iiideteriiiinateness of sign is thus removed. 16. As an instance of the mode of using the formulae, take the problem of finding the locus of a point such that its distance from a given point is in a given ratio to its distance from a given line. We take (rt, 1:) as the coordinates of the given point, aiid it is convenient to take (.r, 3/) as the coordinates of the variable point, the locus of which is required: it thus ll eoines iieci-.<sai'y to use other letters, 5:. iy (X, Y), for i:urreiit coordinates in the equation of the given line. Suppose this is a line such that its perpendicular distance from the origin is =1», and that the inclination of); to the axis of .r is -: 0.; the equation is X cos o.+ Y sin a -1) =0. "[11 the result obtained in§ 15, writing (.r, _l/) in place of (u, L), it appears that the perpendicular distance of this line from the point (.r, 3/) is
- =12—.7: cos a.—]/ sin a;
hence the equation of the locus is '(:c— (()"—l-(1/-Tb)‘-'=c (p - .7: cos a — y sin a), or say (.::—- a)'3+ (y — Ix)‘-’ — c9 (.7: cos a +3; sin a—p)"= , an equation of the second order. The C’0m'cs (Przmbulrr, Ellipse, II_z/pcrbola). 17. The conics or, as they were called, conic sections were migiiially defined as the sections of a right circular cone ; but -pollonius substituted a definition, which is in fact that of the last example: the curve is the locus of a point such that its distance from a given point (called the focus) is in a given ratio , y to its distance from a given line (called the directi-ix) ; taking the ratio as c : 1, then e is called the eccen- tricity. Take FD for the perpendicular t'i'oni the focus F upon the di- l rectri.'. and the given ratio being D 0 F '73 that of c : 1 (c >, -= or < 1, but positive). and let the distance FD be divided at 0 in the given ratio, say we have Ol)=m . 0 UF=em., where m is positive ;—, Fig‘ 1" then the origin_ may be taken at O, the axis 09: being in the direc- tion 01'‘ (that is from O to F), and the axis Oy at right angles to it. l‘_he distance of the poi_iit_(.7_-, y) from F is =»y/(;p._g-my-'+_,):’ its distance from the direetrix is =.L'+m; the equation therefore is (.'/c— cm)"+y'-’=c‘-’ (.1:+/)1)? ; or, what is the same thing, it is (1 ‘ 82)-732 - 22710 (1 +c).z'+ 3/"=0. lf c‘-’=-1, or, since c is taken to be positive, if c=1, this is 3/2 — 4m.i:= 0, which is the parabola. l f c‘-’ not = 1, then the equation may be written / 0 ti I’) (1 _ 62, J. _ mo )- 2: m-e-(1+r') ' K 1-0 +1’ 1—r; GEOMETRY 413 Supposing e positive and < 1, then, writing nt=a(—1e:—'“:), the equation becomes (1 - c‘-’)(x— a>2+ y =a2<1 — as), that is, (ac — «>2 y’ = 1. (L2 a"(1 — ca) ’ or, changing the origin and writing b”=a'-’(1 c”), this is £152 7 3 "‘.; + J.‘ = 19 u- L‘ which is the ellipse. And similarly if c be positive and > 1, then writing 7)L=‘3£e_— 1), e the equation becomes , (1 — e"’)(96+a)9+3/”=a2t1 —c'-’). that ls, (:z:+a_)‘-' _1/3 a“ +2?-'(1’—7) _ ’ or changing the origin and writing 5’=a‘-’(c‘-’ -1), this is '— ‘: = 11 a‘ Ir which is the liyperbola. 18. The general equation a.c2 + :2Iz.'c_I/ + by‘-’ + 27}; + 2g..- + c = O, or as it is written (a, b, c, f, g, h)(.z', 3/, l)‘3= 0, may be such that the quadric function breaks up into factors, = (a.:: +,8_1/ + -y)(a’.L° + By + -y’) ; and in this case the equation represents a pair of lines, or (it may be) two coincident lines. 'hen it does not so break up, the function can be put in the form ){ (ac — (1')? + ([1 — 7/)2 — c‘-’(.v coso. + 3/ sina — p)‘-’}, or, equating the two expressioiis, there will be six equa- tions for the determination of A, a’, 1/, 0,1), (1 , and by what precedes, if ta’, 1:’, (7,1), a are real, the curve is either a parabola, ellipse, or hyperbola. The original coetticients (ti, (2, c, f, [1, ll) may be such as not to give any system of real values for a’ 1/, 0,12, a ; but when this is so the eq11a- tion (rs, I), c, f, _r/, l1)(;r, _1/, 1)? = 0 does not represent a real curve‘; the imaginary curve which it represents is, how- ever, regarded as a conic. Disregarding the special cases of the pair of lines and the twiee repeated line, it thus appears that the only real curves represented by the general equation ((1, b, 0,], _r/, }z)(.r, 3/, 1)‘-’= O are the para- bola, the ellipse, and the hyperhola. The circle is con- sidered as a particular case of the ellipse. The same result is obtained by transforming the equation (n, I), c,_f, 17, Ii)(.c, 3/, l)‘3 = O to new axes. If in the first place the origin be unaltered, then the directions of the new (rect- angular) axes ().r,, 03/, can l)e found so that la, (the co- eflicient of the term .7:,_i/1) shall be = 0 ; when this is done, then either one of the coefiicients of 3:19, 3/12 is = 0, and the curve is then a parabola, or neither of these coefficients is = O, and the curve is then an ellipse or liyperbola, according as the two coefticients are of the same sign or of opposite signs. 19. The curves can be at once traced from their equa- tions :— 3/‘3= -lmx, for the parabola (fig. 13), + = 1, for the ellipse (fig. 14), -1, for the hyperbola (fig. 15) ; and it will be noticed how the foriu of the last equation ‘” =i1 of the (L b Referred to the asymptotes (as a set of puts in evidence the two asyniptotes hyperbola. 1 It is proper to remark that, when (a, b, c, f, g, Ii) (rt, 3;, 1)‘-‘=0 docs represent a real curve, there are in fact four systems of values of a’, I)’, c, p, a, two real, the other two iinaginary; we have thus two real equations and two imaginary equations, each of t.liein of the form (.7: —- u')"’+ (_7/ — b’)'3=e'-‘ (cos a+_7/ cos 3 —p)3, representing each of them one and the same real curve. This is consistent with the assertion of the text that the real curve is in every case represented by a real
equation of this form.