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ORIGIN AND GENERAL EQUATION OF

THE CONIC SECTIONS.

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14. Let a right cone BCD be cut by any plane AMP; required the equation of the curve MAm, which results from the intersection.

B If through the summit B we cause to pass a plane BCD, perpendicular to the base of the cone, and to the cutting plane AMP, the intersection of these two planes will be a right line Aa; and if we

T
cut the cone parallel to its base by
a plane FMG, we shall obtain a
circle whose plane will be perpen-

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A

K dicular to the triangle BCD, and whose intersection with the plane ci AMP will be a right line PM, perpendicular to the two right lines Aa, FG. The line PM will be an ordinate common to both the circle and the section MA».

This premised, call AP=X, PM=y, AB=c, the angle ABa =B, the angle BA a=A; the well known property of the circle gives yy=FPX PG. To find analytical expressions for the lines FP and PG, draw AE parallel to Có, and PK parallel to BD, Loth in the plane BCD; this gives AB : sin AEB::AE : sin B. But

1809-B the L AEB=

90 — 4 B; therefore sin AEB = sin

2 (90—4 B)=cos į B, and consequently from the above proportion

AE =

cx sin B

cos B Again the triangle APK gives_sin AKP : sin APK, or sin AEB : sin AaE, or rather cos B : sin (A+B):: *: AK (because the external angle AaE = the two internal angles A

sin (A+B) and B). Hence AK =

; and therefore KE or

cos & B c sin B-X sin (A+B)

cos & B To find an expression for the part FP; in the triangle APF we have sin AFP, or sin BFG, or sin BGF, or...cos? B :*::sin A:

PG=

FP =

Cos B; therefore by substituting these values of FP, PG,

{

B~zx sin (A+B)}

Cr sin

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x sin A
cos

sin A we have yy=

cos’ | B for the equation required. 15. Now there can occur but

B three cases, Io. That A+B=180°, in which case the cutting plane is parallel to the side BD. This conic section is called a parabola, and its equation is sin A x sin B sin ?B

\P yy = cosB

cos ? B Cx=4 cx sin ?} B, by (12 Fig.) (because sin 18°0=0, and sin A= sin B), or y=+ 2 sin. B.cr. Consequently the parabola is a curve formed by two equal and similar branches, infinitely extended, and constantly receding from each other.

16. II°. If A+B is less than 180°. it is easy to perceive that

B the plane AMP, if continued, must meet the other side BD. The conic section which results is called an Ellipse, and is a curve formed by two equal, similar, and FA finite branches AMa, Ama. Its equation is

sin A yy= cxsin B--xx(sinA+-B)

K 17. III9. If A+B is more than 180°, the section is called an hyperbola, and its equation is

sin A yy=

cos? Į B If we conceive a cone Bcd equal, and opposite at the vertex to the right cone BCD, it is evident that the

d cutting plane AMP, if prolonged, will meet it, and that from their intersection will result a curve M'am' equal, similar, and opposite to the lower curve MAm; or rather these two curves which are called opposite hyperbolas, will form but one same curve represented generally by the same equation.

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cosito

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{ cx sin B+xx sin (A+B—1500)}

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18. Instead of supposing these sections to be made in a right cone, we might have supposed them to have been made in an oblique cone, such as would have been the cone BCD (see figure to 14) if the angle C had not been equal to the angle D. In this case we should have found for the general equation

sin A yy =

sin C sin D This equation, in common with the foregoing one, belongs to an ellipse, a parabola, or an hyperbola, according as the sum of the angles A and B is less, equal to, or greater than 180o.

In the first case it expresses a circle, whenever the angle A is equal to either of the angles C or D; for then we have one or other of the following equations cx sin B

cx sin B
y=-

or....y
sin
C

sin D which are evidently equations of the circle* This circumstance shews that in an oblique cone we may form circular sections in two different manners; one by cutting the cone parallel to its base, the other by cutting it by a plane which makes with one of the sides of the triangle through the axis, an angle equal to that which the other side of the same triangle forms with its base. This is called a subcontrary section.

In the third case, in which the equation expresses an hyperbola, we may suppose c=0, and then we have

sin A sin (A+B_180°) y'=

sin C sin D Suppose the co-efficient of pe equal to a constant quantity, as TY we then find y= which is an equation to a right line; hence

6 we may conclude, which indeed is sufficiently evident, that the hy. perbola degenerates into a triangle, when c=0, that is, when the cutting plane passes through the vertex of the cone.

To reduce the equations of the sections of an oblique cone to those of a right cone, we need only observe that sin C sin Dcos' } B.

Such is the origin of the three conic sections ; we now proceed to investigate their principal properties, and to simplify this analysis, we shall suppose these curves described on a plane.

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This will appear by comparing these equations with the equation of the circle in article 4, which is yy=2 ax-xx; for we have only to make the constant co-efficient

csin B c sin B

-2a and tae eqnations become iden tical.

or

sin C

sin D

OF THE PARABOLA.

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19. The equation of this curve is yy=4 cx sin } B; therefore is we call the constant quantity 4 c sin? | B=p, we shall have

yy=px Hence the squares of the ordinates of the parabola are to each other as their abscissæ. Therefore if we take one abscissa,

EH IG double another, the squares on the

Aln two corresponding ordinates will be in the ratio of 2 to 1. The indefinite line AL is called

Q the axis of the parabola, the point A is its vertex, AQ is an abscissa, MQ the corresponding ordinate to this

P abscissa, and the constant quantity p is called the parameter, or latus rectum, of the axis.

I We can always determine this quantity p by the equation yy=px, which gives x:y::y:p. For this purpose it is sufficient to take any abscissa and its corresponding ordinate and a third proportional to these two lines will be the parameter of the parabola passing through their extremities.

20. If we take the abscissa AF=* p, the point F is called the focus, and the ordinate DF passing through this point will have for its expression, V 4 p=, P. Therefore the double ordinate Dd passing through the focus is equal to the parameler.

If on LA continued, we take AG=AF=ł p, and if through the point G we draw the indefinite line EGe parallel to the ordinate MQ, this line EGe is called the directrir.

p =*+p= AQ+AG = MH, therefore FM-MH; hence the distance from any point M of the parabola to the directrix, is equal to the distance fron; this same point to the focus F. This property suggests an easy method of describing the parabola by a continuous movement, as will be shewn at the end of the conic sections.

22. Let it now be proposed to draw a tangent MT to a given point M of the parabola.

If we suppose the arc Mm infinitely small, its continuation MmT will be the tangent required. Now if we draw the perpendiculars MQ, mq on the directrix, and the lines MF, mF to the focus F, and also mg parallel to Qg, and if from the point F as a center, and with the radius Fm, we suppose to be described the infinitely small arc mr,

which

may
be considered as

a sine, we shall

have MQ--MF, mq=mF; and therefore MQ-mq, or Mg=MF-mF, or Mr. The right-angled triangles Mmg, Mmr are therefore equal and similar, and consequently the angle mMr or TMF=gMm=

QI

G QMT=MTF. Hence the triangle MTF

A is isosceles, and consequently if we take FT-FM, the line MT drawn through the points T and M will be the tangent required.

M The angle MTF = LMO = FMT.

L Therefore all luminous or sonorous rays OM parallel to the axis AP, must on their impinging on the parabola AM be reflected to the focus F; since we know that the angle of reflexion is equal to the angle of incidence.

23. Since FM=5+p, we have FT- P=AT=x. Therefore PT, or the subtangent=21. Consequently in the parabola the subtangent is always double the abscissa.

The tangent MT=V (pr+4 xx)=V (4 MF xx). If we draw the line MN perpendicular to the parabola, or which is the same thing, to its tangent MT at the point M, we shall have PN-PM px

Therefore in the parabola the subnormal" PT 21 is always equal to the semi-parameter. As for the normal MN its expression is (pr+ip)=(MFxp). It is therefore a mean proportional between the distance from the point M to the focus, and the parameter.

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IG 24. Any line MO parallel to the axis of a parabola, is called in general a diameter. The point M is its

Ba

M vertex; four times the distance

a from this point to the focus F is its parameter q; its ordinates are right

P lines NP parallel to the tangent at M and the abscissæ of these ordi. nates are the lines MP.

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• Note. In any curve AM, if TM is a tangent to the curve at the point M and AP and PM co-ordinates to the same point; then TP is called the subtangent , MN perpendicular to TM is called the normal; and NP, the subnormal.

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