88 
CONIC SECTIONS. 
totes to thefe hyperbolas. Cor. z. The afymptotes make 
equal angles with the axis major. 
Prop, to. If the ordinate PN to the axis, fig. 2.7, be 
produced to meet the afymptotes in Q and q, then PQ x 
V q — A a*.—For, by fimilar triangles C A c, CNQ, 
C N ' : C A ! :: NQ ‘ : A. a" \ 
but CA* : CN 2 — CA 2 :: Ait 2 : PN 2 (prop. 7. cor. 3.) 
.-. CN’ : CN ! — CA 2 :: NQ 2 : PN 2 
div. C N 2 : C A 2 :: N Q 2 : NQ 2 — PN 2 _ 
::NQ 2 :iN<Q + PN X n ^ — PN 
:: N Q 2 : P<7 X PQ 
From which, and the fiilt proportion, we have ¥q X 
PQ — A <2 2 . 
Cor. 1. For the fame reafon, j>Q X H = Ai J , or 
Aiz 2 ; therefore P Q X ¥q =■ p q x p Q. Cor. 1. Hence, 
if WVtiaibe drawn parallel to Q q, VW x Vw — Ac 2 ; 
therefore VW x V» = PQ x P q. 
Prop. 11. If a line X Vp x , fig. 28, be drawn in an) 
pofition, cutting the two afymptotes in X x , and the 
curve in V, p. then will VX X V x — pX x P —F°r, 
through V and P draw two lines W Vaw, QP7, perpen¬ 
dicular to the axis, and through P draw R Pr parallel to 
X.v; then, on account of the parallel lines W w, Q q, 
and X x, R r, the triangles Q PR and XVW, Pq r and 
Vxw are fimilar, therefore 
VX:VW::PR:PQ 
V v : V w :: P : P q 
.-. V X X V .v : V W X Vw:: PR x’Pr : PQX P'7; 
but (cor. 2 prop. 10.) V W x V w = P Q x P <1 ; there¬ 
fore VXxV .v=P R X PConceive now the line RPr 
to move parallel to itfelf, until it becomes a tangent 
X. E M at the point E, then P R becomes E L, and P r 
becomes E'M ; hence, VXxV*-ELxE M. For 
the fame reafqn, pX x P x z=.E M x EL; therefore 
VX X Vx =z p Xxp x. 
Cor. 1. Hence , VX—px . For VXx V x— pXx p x 
that is, VXxVj)-f/ue:/iiX V/ij-VX; take V X 
X p x from both fides, and we have VX x Vp — pxxVp, 
tlierefore VX—px. Cor. 2. Hence, if we fuppofe X* 
to move parallel to' itfelf and coincide with LM, we 
fiiall have LE = EM. Cor. 3. Hence, VXxVr=EL'. 
Cor. 4. As VX — px, therefore Xpz=V x ; confequently 
. V X X Xp — E L 2 . Cor. j. Let X x cut C E produced 
in O ; then as the lines L M, X x, are parallel, and L K 
= EM, we have X O — x O ; but (cor. 1.) V X — p x i 
therefore V 0 =j( 0 ; hence, the diameter C E Z bi- 
fedls all its ordinates. 
Prop. 12. If BCb be drawn parallel to L M, and from 
E, E B, E b, be drawn parallel to C M, C L reflec¬ 
tively, then B C b C — L E — M E.—For as CM is 
parallel to B E, and B C to E M, C B E M is a parallel¬ 
ogram, therefore BCrEM. For the fame reafon, 
b C = E L ; but ELrEM; therefore B C = b C =2 
LE — ME. 
Cor. 1. As (cor. 3. prop, n.) V X x V x — E L% 
therefore VXx V x — B C 2 . Cor. 2. As C a is paral- 
Jel to B E, and BC = iC, therefore E a — ab. Def. 11. 
B b is called a conjugate diameter to the'diameter CEZ.' 
Thefe are alfo called conjugate diameters. 
Prop. 13. If C E be produced to meet the oppofite 
hyperbola in D, then D OxE O : O V 2 :: C E 2 : C B 2 . 
For by fimilar triangles CEL, COX, C E : E L :: 
*. E L x CO _ 
CO: OX = — c- - —Ox (prop. 11. 
C E 
hence, VX=OX—OV 
ELxCO — CExOV 
and 
ELxCO 
+ O V 
prop 
C B 
— C B ) 
— C 
fore 
CB 2 
CL 
EL xCO + CExOV 
“ CE 
EL 2 x CO 2 —CE 2 xOV 2 
cor. 5.); 
= —CE-° V = 
E LxCO 
Vx = Ox+OV^ -- 
therefore (cor. 1. 
12.) 
X C E 
E 
C 
CL 2 
: (as E L 
C B 2 x C O 2 
X O V 2 , there- 
E 2 X O V e — 
= CB 2 ; hence, 
x >J 2 —■c.L 2 = CB 2 
X C u + C L x CO — c E 
— C B 2 x CO +LD (DO) 
X EO ; hence, DOxEO; 
OV :: CE 2 : CB 2 . 
Prop. i+. If D V, SE, 
fig. 29, be drawn parallel to 
the afymptote Y C, ’and 
ET parallel to C Z, tlien 
DVX D C—S Ex S C.—For 
draw V R parallel to CZ, 
then the triangles XVD, 
LES, V.vR, EMT, being 
fimilar we have 
DV: SE:: VX: LE 
VR(DC) ET (SC) 
V.v: EM (LE) 
D V X DC: SE x SC:: VXx Vx:LE 2 ; 
but (cor. 3. prop. 11.) VXxVx— I.E 2 ; therefore 
D VX DC = SE x SC. 
Cor. 1. B e caufe LE r E M (cor. 2. prop. 11.) and 
S E is parallel to CM, tlierefore LS —SC, or. S C — 
|CL. Cor. 2. Hence, DVxhCisa conftant quantity, 
and therefore DV varies inverl'ely as DC. Cor. 3. If the 
point 
