36 
CONIC SECTIONS. 
And, if S/; •— Hp — H P — S P, the hyperbola generated 
by the two lines S p, Up, revolving about S and H, is an 
opposite hyperbola. 2. The points H, S, are the foci. 
3. If A and M be the points where the curves interfecl 
the right line HS, the line AM is the axis major. 4. 
Let AM be bifefted in C, the center, and BCN be 
drawn perpendicular to AM; then, if from A as a cen¬ 
ter, with a radius S C, a circle be defcribed cutting B N 
in B and N, B N is the axis minor. And when AM = 
BN, the hyperbola is called equilateral. 5. If, in like 
manner, two hyperbolas d. Be, q Nr, be defcribed about 
the two ax.es BN, AM, they are conjugate hyperbolas. 
6. A right lineLST drawn perpendicular to HS, and termi¬ 
nated by the curve, is the parameter, orlatus redtum. 7. 
If PCG be a right line palling through the center, it is a 
diameter. 8. If QV be drawn parallel to a tangent at 
P meeting C P produced in V, then QV is an ordinate, and 
GV, PV, are abfeiflas. 9. The afymptote is a right 
line, which continually approaching the curve, arrives 
nearer to it than by any allignable quantity ; but, if in¬ 
definitely produced, never meets it. Cor. Hence we may 
confider the afymptote as the limit to which the tangent 
approaches, when the $.bfcilfa is increafed without limit. 
10. If T V, in fig. 2i, 
be a tangent atT, meet¬ 
ing the axis at A M in 
V, and the line xVy 
be drawm perpendicu¬ 
lar to AM, that line 
is the directrix. 
Prop. i. The dif¬ 
ference of two lines 
drawn from the foci to 
any point of the curve, 
is equal to the major 
axis.—Per, conceiving the deferibing points P, p, fig. 20. 
to come to A and M, we have HA —-AS = 1 IP — PS 
= S p —pH —SM—• MH, take from the firft and laid 
of thefe, the common part MA, and we have MH — 
AS —AS—MH, therefore 2MH = 2 AS, and MH 
— A S ; hence, HP — PS-HA —.AS =HA — MH 
= M A, or 2 AC. 
Cor. 1. Hence, SC — CH. Cor. 2. By tranfpofition, 
HP = 2AC a.SP. 
Prop. 2. The rectangle under the didances of either 
of the foci from the twb vertices, is equal to the fquare 
of the femi-axis minor.'—For B C 5 = B A : — AC S = 
(b y conflrudtio n) SC 2 — AC2 = b(J 4- AC X SC— AC 
= SC q- Civl x S A — SM x S A. 
Prop. 3. The latus reftum is a third proportional to 
the axis major and the axis minor.—By cor. 2. prop. 1. 
HL= 2AC4-LS; hence, HS 2 4- S L 2 ( = HL 2 ) = 
4'A C* 4- 4 A C x L S L S z , therefore FI S 2 — 4 A C 2 
= 4AC x LS; but HS = 2CS = (by confiruftion) 
a A B ; lienee, 4AB 2 —4AC 2 = 4AC x LS, or 4B C 2 
= 4 A C X L S ; therefore 2 AC: 2BC:: 2BC ; :LS, 
that is, AM : BN :: BN : LT. 
Prop. 4. If P V, fig. 22, bifeft the angle S PH, it will 
be a tangent to the curve at P.—Fpr, take P??z = PS, 
and join S m ; 
then, as m P = 
P S, z m P n — 
Z SP»;, and P « 
is common totlie 
triangles mV n, 
SP«, we have 
mn — S n, and 
the angles at 
n right angles 
fEnc.b.i. p.4.) 
Now, if PV be 
not a tangent, 
let it cut the curve at forne point Q, and join QS, Q m. 
4 * 
Then, as mn = nS, and the angles at n are right angles, 
and nQ common to the triangles ?nnQ_, SnQ, we have 
»zQ=SQ. Now, (prop. 1.) MA = HP—-PS=:HP 
—■ P m = H»i; alfo, H m 4- m Q is greater than HQ (Euc. 
b. i. p. 20.) ; therefore Hor M A, is greater than H Q 
— in Q, or greater titan HQ — Q S, confequently Q is not 
a point in the curve (prop. 1.); therefore, as PV cannot 
meet the curve at any other point, it muff be a tangent. 
Cor. 1. Hence, two right lines drawn from the foci to 
any point of the curve, make equal angles with the 
curve. Cor. 2. IfSP, fig. 23, lae produced to meet a, 
line C E parallel to the 
tangent V P at P in E, 
the part PE is equal to 
AC. For, draw HI pa¬ 
rallel. to C E, meeting 
S P produced in I. Then, 
as PV is parallel to I H, 
the ZSPV=/PIH, 
and/HPV = /PHI; 
but, as the Z^PV = 
/HPV, the Z PIH = 
Z PH I, therefore PH = 
PI. Alfo, as SC = CH, 
and C E is parallel to 
HI, SE’= IE, or IE = 
| S I; and, as PI = P H, 
PI = 4 (PH 4- PI) ; 
therefore PE = PI — 
IE = 4 (PH 4-PI — SI) =4 (PH + PI— PS — PI) 
— a (PH— PS)=AC (prop. 1.) 
Prop. 5. If two perpendiculars be drawn from the 
foci to a tangent at any point of the hyperbola, they will 
meet it in the periphery of a circle defcribed on the ma¬ 
jor axis as a diameter.—For produce the perpendicular 
S Y, fig. 24, on the tangent PV, to meet HP in a ; then 
thez YP«2 = Z YPS 
(prop. 4.), Z SYP = 
Z a Y P, and P Y is 
common to the two 
triangles S PY, zzPY, 
therefore S Y = Y a, 
and a P = S P ; hence, 
Ha ( — HP —'Pa) = 
HP — SP = (prop. 1.) 
AM. Now, as SC = 
CH, and SY = Ya, 
CY is parallel to Ha; 
therefore SH : SC :: 
Ha (AM) : CY; but 
SC = ASH; there¬ 
fore CY = |AM = AC; hence, Y is in the periphery, 
of a circle, whofe center is C and radius CA. For the 
lame reafon, if P S be produced to meet H Z produced 
in b, CZ — | Si = (as H£ is parallel to Sa) |Ha = AC 
as before. 
Prop. 6 . The redtangle under the perpendiculars from 
the foci upon the tangent, is equal to the fquare of the 
femi-axis minor.—Let H Z interfeft the circle in p, and 
join Cp; then the ZpZY being a right angle, it mull 
be in a femicircle, as Z and Y (prop. 5.) are in the 
circumference ; confequently p CY mud be a diameter; 
hence, in the triangles HC p, SCY, HC21SC, C p — 
CY, and the Z HC/? — Z SCY, therefore Hj)=:SY ; 
and, by the property of the circle, HP x HZ', or § Y X 
HZ, = HA x HM = BC» (prop. 2.) 
Cor. 1. Since (prop. 4.) ZSPY = ZHPZ, the tri¬ 
angles SPY, HPZ, are fimilar, and SP : S Y :: HP : HZ 
- SY X H1> - hence ,iilXHP = H ZX SY=BC. i 
SP 
SP 
and S Y 5 = B C ! X 
ir 
SP 
Cor. 2. As BC is conftant, 
SY 
