OF THE PARABOLA. 19. The equation of this curve is yy-4 cx sin2 B; therefore it 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, double another, the squares on the two corresponding ordinates will be in the ratio of 2 to 1. The indefinite line AL is called the axis of the parabola, the point A is its vertex, AQ is an abscissa, MQ the corresponding ordinate to this abscissa, and the constant quantity p is called the parameter, or latus rectum, of the axis. N EH IG A 12 R Fd M L We can always determine this quantity p by the equation yy=px, which gives xy::yp. 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 pp. Therefore the double ordinate Dd passing through the focus is equal to the parameter. If on LA continued, we take AG-AF= p, and if through the point G we draw the indefinite line EGe parallel to the ordinate MQ, this line EGe is called the directrix. 21. Now FM= √ {yy+(x− + p )' } = √ { px + (x− + p )} =x+&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 from 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 Qq, 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= QMT MTF. Hence the triangle MTF is isosceles, and consequently if we take FT FM, the line MT drawn through the points T and M will be the tangent required. The angle MTF = LMO – FMT. 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+p, we have FT-p-AT-x. Therefore PT, or the subtangent 2. Consequently in the parabola the subtangent is always double the abscissa. The tangent MT=√ (px+4 xx)=√ (4 MF × x ). 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 Pr = Therefore in the parabola the subnormal PT 2 x is always equal to the semi-parameter. As for the normal MN its expression is✓ (px+4 p2)=√ (MF xp). It is therefore a mean proportional between the distance from the point M to the focus, and the parameter. 24. Any line MO parallel to the axis of a parabola, is called in general a diameter. The point M is its vertex; four times the distance from this point to the focus F is its parameter q; its ordinates are right lines NP parallel to the tangent at M and the abscissæ of these ordinates are the lines MP. EH M IG A 12 D R N L 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. T M A PN To find the equation of the co-ordinates of the diameter MO, call MP, '; PN, y'; AQ=AT=a; we shall have MQ=√ ap, q=p+4a, MT=aq; and if we draw NL perpendicular to the axis, the similar triangles NRL, MTQ, will give NL MT: MR :: MQ: or✅aq: y' +√ ag:: ✔ap: NL_√ap ag ✔aq 2 ay' But AR RT-AT-x-a; therefore AL-x2+a+ and ✓ aq by the property of the parabola, NL2px AL, or (√ ap + √ up)2=ap+px' + 2 apy ; from which by reduction we obtain y'y' qu', an equation similar to that which was before found for the co-ordinates of the axis. Hence we may conclude that any diameter MO bisects all the ordinates Nn. The two following problems afford an easy application of the above principles. 25. I. Given the axis AL of a parabola and its parameter p, to find a diameter MO, which shall make with its ordinates a given angle MPA. (See preceding figure). This problem merely requires us to find the point Q where the perpendicular MQ meets the axis. For this purpose call AQ=a, the triangle MTQ will give AT: MQ:: Radius; tang A (because MPn=/ MTQ) or in symbols, 2 r: √pr::1: tang A Hence x = P = 4 tang A p cot 'A, and the parameter of the dia meter MO, or q=p+4x=p (1+cot A)=p cosec A sin A It is easy to see that this problem has two solutions. Prob. II. Given the parameter q of the diameter MO, with the vertex M of that diameter, and the angle A which it makes with its ordinates; to find the axis AL, its vertex A, and its parameter p. (See preceding figure.) This problem requires us to find the distance MQ from the axis to the diameter, and then the distance AQ, in order to obtain the vertex A and the parameter p. Preserving the same denominations as in the preceding problem, we have MQ = √ pr, q=p+4x= Hence we obtain p=q sin oA, x= q cos 2A, 2 sin2 A and MQ=±} q sin A cos A+ q sin 2 A. The properties of the parabola are of frequent and great use in the Arts and Sciences. to the transverse axis Aa. Call this transverse axis 2 a, and we shall have yy= sin A sin (A+B) (2 ax—xx) cos2 B 26. The double ordinate BCb passing through the middle C of the axis Aa, or through the center of the ellipse is called the less or conjugate axis. To introduce it into the equation of the ellipse, call it 2 b; and since when y=b, x=a, we shall have sin A sin (A+B) aa. Hence by substitution we find bb bb cos B yy=— (2 ax―—xx) aa This expression gives the following proportion yy 2 ax-xx::bb: a2 or PM: AP× Pa::CB2: CA; that is, in the ellipse the squares of the ordinates of the transverse axis are to the products of their corresponding abscissæ, as the square of the conjugate or lesser axis is to the square of the transverse axis. If we describe a circle whose center is C and radius CA, we shall have PN=AP x Pa. Therefore PN: PM::a;b:: CB'; CB Hence the ordinates of the ellipse are proportional to the ordinates of a circle described on the transverse axis; this gives us an easy method of describing an ellipse. It is only necessary for this purpose to describe a curve through a series of points taken on the ordinates of a circle cut into similar parts. 27. If we had reckoned the abscissæ from the center C, making CP-2, we have x-a-z, and substituting this value in the equation above found, we have bb yy==(aa-zz ) — b2 refer. aa b2 22 -; an equation to which we shall often If b were equal to a, we should have yy-aa-zz an equation of the circle; hence we may consider the circle as an ellipse whose two axes are equal. zz: bb-yy:: aa bb, or MQ; BQ × Qb :: CA2 : CB2. Therefore in the ellipse the squares of the ordinates of the lesser axis are to the rectangles of their abscissæ as the square of the transverse is to the square of the conjugate axis. Hence the semi-conjugate axis is a mean proportional between the distances from either of the foci to the two vertices of the ellipse. 29. The ordinate DF passing through the focus has for its expression -; and Dd, which is called the parameter p of the trans 62 a verse axis= 2bb 4bb a 2 a Consequently 2 a 2b 2b: p, and therefore the parameter is a third proportional to the transverse and conjugate axis. By analogy to this property we call the parameter |