81 
CONIC SECT! O N S. 
»f the lines drawn from S and H equal to H A -f A S — 
SH+iA'S, and in the latter cafe, the fum is equal to 
S H + 2 HvM ; hence (definition i.), S H 2 A S — S H -|- 
aHM, confequently AS —HM; hence, HP + PS = 
HA + AS = HA + HM-AM, or—2AC. 
Cor. 1. Hence, SC — CH. Cor. 2. By tranfpofition, 
IIP-2AC-SP. 
Prop. 2.—A line drawn from either of the foci to the 
extremity of the axis minor, is equal to the fe mi-axis 
major.—For the triangles HC B, SCB, have SC —CH 
fcor. 1. propolition 1.), BC common, and the angle S C B 
— the angle HC B, therefore (Euc.b. i. p. 4.) SB = BH; 
but (prop. 1.) SB + BH—2AC ; therefore 2AC22: 
2 S B ; confequently AC — SB 22 BH. 
Cor. Hence, M S x S A = B C 2 . ForBC 2 — BS 2 — 
SC 2 =r AC 2 —SC 2 = AC + SC'X AC —SC —MC-I-SC X 
AC — SC = MS’x S A. 
Prop. 3.—The latus reflum is a third proportional to 
the axis major and the axis minor.—By cor. 2 prop. 1. 
LH-2AC-SL; hence, HS 2 -fS I.a— L H 2 =4 A C 2 
— 4ACXSL + SL 2 , therefore HS 2 4 4AC X SL — 
4 A C 2— (prop. 2.) 4S B 2 ; hence, 4ACXSL-4SB 2 
— HS 2 — (becaufe 2 SC=HS) 4SB 2 — 4SC 2 —4 B C 2 , 
therefore 2AC:2BC::2BC:2SL, or AM : B D :: 
J3 D: LT. 
Prop. 4. II SP, fig. 2 1 be produced to m, and the 
right line VPWbi- 
fect the angle HPw, 
it will be a tangent 
to the eliipfe at the 
point P.—For, take 
PmPH, and join 
H in ; then as m P — 
PH,Z m P n — X HP;;, 
and P it common to 
tire triangles m P 11, 
HPe, we have inn — 
H n, and the angles 
at n right angles ( Euc. 
b. i. p. 4. and d. 10.), 
Now if PW be not a tangent, let it cut the curve at forne 
point Q, and join QH, Qm, QS. Then as Mn = nil, 
the angles at n right angles, and n Q common to the tri- 
atigles mn Q, H?zQ, we have »Q —HQ. Now (Euc. b. 
i. p. 20.) S QQ?» is greater than S in, or greater than 
5 P + P m, or S P 4- P H, or A M (prop. 1.); therefore 
SQ-fQH is greater than AM, confequently Q is not 
a point in the curve (prop, i.); therefore as VPW 
cannot meet the curve at any other point, it mult be a 
tangent. 
Cor. 1. A tangent at A, or M, is perpendicular to A M. 
Cor. 2. As the angle m PW — opp. angle SPY, we have 
the angle HPW — the angle SPY;' that is, two right 
lines drawn from the foci to any point of the Curve, make 
equal angles with the curve. Cor. 3. Complete the pa¬ 
rallelogram SPHG; then SG + GH-SP + PH—AM; 
therefore' (def. 1.) G is a point in the eliipfe ; and as the 
diagonals of a parallelogram bifeft each other, PG paffes 
through the center C of the eliipfe ; hence, the center 
bifeeds all the diameters. Cor. 4. Draw the tangent G t 
at the point G. 
Then the /SPH—- 
XjSGH; but as 
ZSPV-/HPW, 
the Z SPY — f fup- 
plement of X S P H ; 
for th.e fame reafon, 
the Z t G H— f fup- 
plement of Z S G H ; 
hence, z S P V — Z 
tG H ; and as S P is 
parallel to G H, t G 
mult be parallel to 
PV. Cor. 5. The part PE of the line PS, fig. 1 1, inter- 
. -Vul. V. No. 255. 
ccptcd between any point P and the conjugate diame¬ 
ter DCX, is equal to - AC. For draw IT I parallel to 
DCK, and PF perpendicular to it, and confequently 
perpendicular to HI, interfering HI in X. Then as 
Z SPV —x: HPW, (prop. 4. cor. 2.) the / IPX;- 
ZHPX, and PX is common to the two right-angled 
triangles PXI, PXPI; hence, PI = PH. But as SC — 
CH, and CE is parallel to HI, SE —IE, or'IE—->-S 1 
and asPIrrPH, PI—i (Pl-f PH) ; hence, PE — I K 
+ PI rr i (PI + S I + P H) - $ (S P + P H) - A C 
(prop, 1.) 
Prop. 5.—If a circle be deferibed on the major axis 
of an eliipfe, and from each of the foci perpendiculars be 
drawn to a tangent at any point of the eliipfe, they will 
meet it in the periphery of the circle.—For let S Y, H 7 , 
as in fig. 12, be drawn from the two foci perpendicular 
to the tangent Y Z 
at P ; join Y C, pro- ^ q 
duce H P to W, 
making PW — PS, 
and join WY. Then 
(prop. 4.) z w 8Y 
“ZSPY, and as 
PS—PW, and PY 
is common to the tri¬ 
angles SPY, WPY, 
W Y — Y S, and 
ZPYS— ZPYW; 
but the angle PY 
is a right angle ; 
therefore PYW is 
alfo a right angle ; 
confequently SYW 
YW, and (cor. 1 t ._ v ... t _,..- 
to H W; hence, S C: C Y ::SH:HW; but S CriSH y 
therefore C Y— i HW — \ (H P + P S) — (prop. 1.) 
AM —AC ; hence, as C Y —C A, Y mu It always be in 
the periphery of a circle whofe center is C and radius 
CA. In the fame manner it may be proved for the 
other perpendicular HZ, that Z is in the periphery. 
Pr.op. 6. The rectangle under the perpendiculars from 
the foci upon the tangent, is equal to the fquare of the 
femi-axis minor.—For produce Z H to meet the circle in 
E, and join CE; then (prop. 5.) Z being in the peri- 
phery ol the circle, and the angle H Z Y a right angle, 
the angle Y Z E mu ft be in a femicircle ; confequently 
Y C E mu ft be a.diameter. Hence, in the triangles Y CS, 
ECU, YC-EC, SCrHC, and Z Y C S =Z E CH, 
therefore E H — YS; but by the property of the circle, 
ZII x H E — M H x H A — (cor. prop. 2.) BC 2 ; hence, 
Z H X S Y = B C 2 . 
Cor. 1. Becaufe (prob. 4. cor. 2.) Z SPYr:Z hPZ, 
the triangles SPY’, PIPZ are fimilar, and SP : SY :: 
H P : HZ z= Ipl, hence, I™? = S Y x H 2 
K 
is a ftraight line. Now as SY 2 ' — 
prop, i.) S C — C I!, C Y is parallel 
= B C 2 ; therefore SY’ 2 2z:BC 2 x 
is conftant, S Y^ Varies as yj- 
.fs~p 
HP 
Cor. 2. As B C 
prop, i.) 2 A C — S p; therefore S Y T varies as \J 
r 
(cor. 
~sl~ 
or- 
SY 
vanes as y 
hAC-SP 
2 AC — SP 
Cor. 3. Becaufe 4SY 22 
4.BC 2 x SP 
~ 2 A C — S F 
therefore 4SY’ 2 — 
S P 
and (prop. 3.) 4B C ! =; ?AG x I 
2 A C x L x S P . hencCj L x S P 
2 A C — S P 
2 A C x L x S P. 
4SY 2 :: LXSP: ; 2 A C-S P : 2 A C, 
^ 2 A C — S P 
S P is lels than 2 A C, I, x S P is 
and as 2 A C • 
than 4 S Y’ 2 , 
Y 
lefs 
Pro?** 
