CONIC S 
(Tp 
SY varies as \/^i but H P 222 2 A C 4- S P (cor. 2. 
Jri r 
V T r c V • ( ST I 
prop, 1.); therefore S Y vanes- as 3/ 2 A 1 4- a ' B ' or SY 
varies as J -: 1 g — * C ° r • 3 * Since 4 S Y 2 = 
and (P ro P- 3-) 4B C 2 222 2 AC X L, we 
2 nt 4- ^ *- 
1 c -ir f _ 2 A C X Lx S P - T cd. 
have 4SY 2 —— 5 —; hence, Lx SB : 
4SY 2 :: L X SP : 
2 A C + S P 
2 A C x I- x S P 
2 A C 4- S P 
2AC 4- SP : 
2 A C ; and as 2 A C 4- S P is greater than 2 A C, L X 
S P is greater than 4 S Y 2 . 
Prop. 7. If an ordinate be drawn to the major axis of 
an hyperbo¬ 
la, the pro¬ 
duct of the 
abfeiflas is to 
the fquare of 
the ordinate, 
as the fquare 
of the femi- 
axis major to 
the fquare of 
the femi-axis 
minor. Draw 
P N, fig. .25, 
perpendicular to AM, and take AD = PS: then, 
' PS = CD-CA 5 
and PH =2 CD 4- C A (prop. 1.) 
Alfo, NS = NC-CS 
NH = NC + CH222NC 4- CS (prop. 1. cor 1 1 
but PH 2 — PN 2 4- NH 2 (Euc. b. i. p. 47 .) '* 
PS 2 =2 PN 2 4- NS 2 ; that is, 
C D 4- C A 2 = PN 2 4- N c 4- C S 2 
CD — CA 2 — PN 2 4- STc 
• CS 2 
fub. and CD x CA 2= N C x CS, 
c v C S 
therefore C D 2=-^— i hence,. 
P S = C D — C A ^2 ~' C - X ; C - ~ — AC 2= NCx CS—CA 2 
C A 
But PS 2 — PN 2 4-NS 2 ; that is, 
CA 
nc ! h c s 2 
iNCkC'kCa' + C^ 
— PN 1 4.N C*_,N C X CS + CS a 
which is the fame equation as that for the ellipfe in 
prop. 7. and consequently there refults from it the fame 
•concluiion, that is, AN X NM; PN 2 :: AC 2 : BC 2 . 
B C 
Cor. 1. Hence, PN222— x 3/ AN X NM; there¬ 
fore, if upon the fame axis major AM, to any other axis 
minor bg , an hyperbola v A xo be drawn, and p be the 
point where it interfedfs the ordinate N P, thenN* = 
b C __ 
X T A N X N M ; hence, NP : N/) as BC : iC, 
that is, in a given ratio; therefore, for the fame reafon 
as in the parabola, prop. 20. the area APS : area A/>S 
in the fame given ratio of B C : b C. Cor. 2. If L= the 
Jatus reftum, and G B be the axis minor; the., (prop. 
3.) AM : BG :: BG : L; hence, AM: I. :: AM 2 • 
BG 2 :: AC 2 : BC 2 :: AN x NM : PN 2 , therefore 
AM X PN 2 
2A C x PN 2 „ 
AN x NM" Dr ’ 3 - LAC = 
L — I_^_ 
AUxNM 
a, B C =22 b, C N 2= x y N P — y, we have x —. a x a- 4- a 
(x- ;— a-) : y- :: a 2 ; b 2 , therefore y 2 =2: — x x 2 _ a 2 . 
E C T I O N S. 67 
Cor. 4. Becaufe SP 22 CD —.CA, and H P r CD -j- 
C A, therefore 5.SB 4- n P 22: C D ; hence, CN x CS 
(2: CA X C D) —CA 4- | ."SP + H P. Cor. 5. As 
A C and B C are conftant in the fame hyperbola, A N 
X N M varies as P N 2 . 
Prop. 8. If PT be a tangent at P, then will CN : 
C A :: C A : CT.—By prop. 4. PT bifedfs the angles 
SPH; 
.-. HP : SP :: IIT:TS (Euc. b. vi. p. 3.) 
and HP 4- PS : HP —PS :: HT+TS: HT —TS, 
but HP — PS — 2 AC (prop. 1.), and HT — TS — HC 
4- CT—TS — SC 4- CT —sc —CT22 2CT ; hence, 
HP 4 - PS : 2 A C :: HS : 2 C T, 
or AC:.CT::|.hr 4- B - : j- ■ HS , or C S, 
:: A C X 5 • V! B 4 - B o : C S X A C 
:: CS X CN (prop. 7. cor. 4.) : C S x AC 
:: CN : AC, 
hence, CN : C.A :: CA : CT. 
Cor. 1. By divifion, N A : CA :: AT : CT, and as 
C A is greater than CT, NA is greater than AT. Cor. 2. 
If AC — a, B C — b, CN 222 x, N P —y, then x : a :: a : 
C T =2 — ; hence N T — x -22 --; confe* 
x x x 
b 2 _ 
quently (cor. 3. prop. 7.) PN 8 : TN’ :: — Xw z — a* 
^ -■ U 1 , a 
• --- :: b 2 x 2 : a 2 x 2 — a’; 
x 2 
Prop. 9. To draw an afymptote to an hyperbola.—• 
a 2 
By cor. 2. prop. 8. CT = — ; now increafe x without 
limit, and C T, fig. 26, will be diminifhed without limit. 
confequently T approaches C as its limit; therefore, 
(def. 9. cor.) the afymptote palfes through C. Alio 
(cor. 2. prop. 8.), PN 2 : TN 2 :: b 2 x 2 : a 2 x 2 -a 5 , 
the limit of which ratio (by increafing x without limit) 
is b 2 x 2 : a 2 x 2 :: b 2 : a % and P N : T N :: b : a, or, 
as T now coincides with C, P N : C N :: b : a. But, as 
P approaches the afymptote as its limit, and T approaches 
C ak its limit, the limit of the triangle PNT is a tri¬ 
angle fimilar to a A C ; therefore, at that limit, P N : 
CN :: A a: AC (a) ; hence, and the laft proportion, 
we get A a 222 b — be• if, therefore, A a be erected per¬ 
pendicular to A C, and equal to B C, and CaZ be drawn, 
it will be the afymptote. Hence, if on the other fide of 
A C, a perpendicular A b be erefted equal to B C, and 
and C b L be drawn, it will be an afymptote to the other 
part; and if thefe afymptotes be produced on the other 
fide of the center, they will manifeftly-be afymptotes to 
the oppofite hyperbola. 
Cor. 1. Thefe afymptotes will be afymptotes- to the 
conjugate hyperbolas; for perpendiculars to B G, at B 
and G, equal to C A, will terminate at a- and b ; and 
therefore, by the propofition^C a Z, C b~L , will be afymp¬ 
totes 
