CONIC SECTIONS. 
89 
point E, fig. 30, coincide with A, the angle ACT=; 
Z. ACS (cor. 2. prop. 9.) ; but as A T C S is a paral¬ 
lelogram, the / ACT~^ CAS ; therefore the /_ CAS 
A C S ; confequently A S zz S C — A T. Cor. 4. 
Hence-, DVx DC —AS 2 . Cor. 5. The triangles BCS, 
C GT, are fimilar, and BCrCG; therefore G T — C S 
— T A. For the fame reafon, B S = S A. 
Prop. 15. If LMN be drawn parallel to CD, then 
LM=MN.—For (cor. 4. prop. 14.) LMxMC— 
TA’, and NTxMC = TG 2 ; but T A = T G (cor. 5. 
prop. 14.); therefore LM = MN. Cor. Hence (prop. 
12. cor. 2.) NCa is a conjugate, diameter to. the dia¬ 
meter LC^,'and confequently parallel to a tangent at L. 
For the fame reafon, LCLis a conjugate diameter to 
the diameter NC«, and therefore parallel to a tan¬ 
gent at N. 
Prop. 16. All the parellelograms circumfcribing any 
two conjugate diameters of an hyperbola, are equal.— 
Let ale d, fig. 31, be a parallelogram circumfcribing the 
two conjugate diameters D K, P G ; join P D, and draw 
D 1/ perpendicular to the afymptote C Z. Now (prop. 15. 
and its cor.) the afymptote CZ bifetts DP, which is 
the diagonal of the parallelogram CD aP; therefore the 
angle a mult be fituated in that afymptote. A Ho 
(prop. 15.), D m P (>n being the point where D P inter¬ 
feres CZ) is parallel to the afymptote C L, and confe¬ 
quently given in pofition, or the angle Dinn is conftant, 
and therefore D m is to D n in a given ratio ; but (prop. 
14. cor. 2.) D m varies inverfely as Cvr-, therefore D n 
varies inverfely as C« ; but (cor. 1. prop. 14.) Cm — 
Car therefore D n varies inverfely as C a, and confe¬ 
quently D n X C a is content ; but D n X C a is equal 
to the area of the'parallelogram D a PC ; therefore the 
parallelogram DaPC is a content quantity ; but tire 
parallelogram DaPC is a fourth part of the parallel¬ 
ogram a bed j therefore the parallelogram abed is a 
content quantity. 
Cor. \. Hence, the parallelogram abed— the parallel¬ 
ogram v w xy deferibed a out the major and minor axes 
AM, B FI ; therefore their fourth parts, aDCP, AoBC, 
are equal; if therefore P F be perpendicular to C D, 
"Vo 1 \ V-. No. 256. 
we have CDx PF = ACxBC. Cor. 2. Bceaufe the 
triangles SPY, HPZ, FPE, fig. 23, are fimilar, we have 
SP: S Y :: f E : P F 
H P : HZ :: PE : PF 
SP X HP : S Y X H Z :: PL 2 : F F 2 ; 
but (prop. 6.) SYxHZ = BC 2 ; and (cor. 2. prob.4.) 
PE'rrAC* ; therefore S P X H P = ^ 1 X A C -—CD 2 
P F 2 
by the laft corolla. 
Pscqp. 17. The difference of the fquares of any two 
femi-conjugate diameters, is equal to the, difference of 
the fquares of the two femi-ax.es.—Draw PN, fig. 31, 
perpendicular to the axis H S ; then ( Euc. b. ii. p. 12,12.) 
PS 2 =PC ! + CS 2 -2.CSxCN 
PH* = PC 2 +CS 2 (CH 2 ) + 2CS (CM) XCN 
P S 2 + P H 2 = 2 P C 2 + 2 C S 2 j ' 
and 2 P C 2 =-P S 2 + P H 2 — 2 C S 2 ; 
but j C D * = 2 P S x P H (prop. 16. cor. 2 .) 
.•. 2 P C 2 —2CD 2 - PH —■ P S * — 2CS 2 — (prop. 1.) 
4AC 2 — 2CS“— (def. 4.) 4AC 2 — 2AB 2 :=4AC 2 —2AC 1 
— 23 C 2 =. 2 AC 2 — 2BC 2 : hence, PC 2 — CD 1 — AC 2 — 
BC 2 . 
Prop. 18. If A SW, fig. 21, be the axis of the hyper 
bola PAL, the ditence S P 
CM — SC X col. P S W- 
Let PN be an ordinate to M A N ; then (prop. 7.) C M* 
: CB 2 :: AN X NM ; PN 2 ; but CB* = AB 2 — AC*=SC‘ 
— CM 2 ( de f. 4.), a nd AN x NM=lN — c A X 
Cw 4- CM = CM X CN + CM — CN 2 — CM 1 — 
SCqrSfN 2 — CM 2 ; hence, CM 2 : SC 2 —CM 2 ::2C + ai v* 
SC* — CM 2 x SC rpSN 2 
CM 1 
■ CM 2 
— CM 1 : PN — 
EC 4 + SC 1 xSN 2 qr2SC 3 xSN— 2SC*hCM 2 ^2 s x SnkcM ! -CM'- XSN=4-CM 2 
add S N 2 to both fides of the equation, and we have 
S P 2 (PN 2 + SN 2 ) — 
sc'p sc 1 X SN 1 qc 2SC 3 X SN'i-asC X iCM*f±r lSC>< SN * CM 2 + CM 1 
* 
the fquare root of which is S P — SC 2 _|_S\ : x SC. CM 2 
C M 
= (as SC 2 — C M 2 = B C 2 1 b C’ T S N x S C . 
’ C M 
but + SN = SPx cof. P S W ; therefore SP - 
f • + scxsrxco L P 2 w. hence; sp = 
CM 
B C 2 
CM-SCx cof. PSW 
Prop. 19. If a line be drawn from the focus to any 
point of the curve, and from that point a line be drawn 
perpendicular to the diredtrix, thefe lines are i'll a given 
C A 2 
ratio.—For CV — ———(prop. 8.) therefore SVz: 
S C 
SC- 
B C 
C A 
s c 
C A 
A B ■ 
C A 
s c 
! : S P, or 
sc sc sc 
and by trigonometry, rad. = 1 : cof. PSW 
B C 
CM — S C x cof. PSW (P ro F- l8 - ) : SN 
B C 2 X cof. PSW 
;• therefore KP =: VN 
~ C M — S C x cof. PSW 
= &v +.SN = 4 £! + B C 2 X cof. PSW 
s c 
CM x B C 2 
S C’xCM - 
BC 2 
S C X col. PSW 
CM — SC X cof. PSW 
; henpe, S P : P K- :: 
CM x BC 2 
0 m — SC X col'. PSW ‘ SC x CM — SC X- cof, PS.W 
:: SC ; CM.a conftant ratio. 
A a Cor , 
