CONIC SECTIONS. There are three curves, whose properties are extensively applied in mathematical investigations, which, being the sections of a cone made by a plane in different positions, as will be shown in another place, are called the Conic Sections. These are, 1. The Parabola. 2. The Ellipse. 3. The Hyperbola. PARABOLA. DEFINITIONS. 1. A Parabola is a plane curve, such, that if from any point in the curve two straight lines be drawn; one to a given fixed point, the other perpendicular to a straight line given in position: these two straight lines will always be equal to one another. 2. The given fixed point is called the focus of the parabola. 3. The straight line given in position, is called the directrix of the parabola. Thus, let QAq be a parabola, S the focus, Nn the directrix ; Take any number of points, P,, P,, P, ... in the curve; 39 N Ni P N2 31 Join S, P, ; S, P, ; S, P ̧'; ... and draw P, N,, P, N,, P, N,, ... perpendicular to the directrix; then 39.... SP,=P, N1, SP,=P,N,, SP ̧= P,N,,.... N3 R 4. A straight line drawn perpendicular to the directrix, and cutting the curve, is called a diameter; and the point in which it cuts the curve is called the vertex of the diameter. 5. The diameter which passes through the focus is called the axis, and the point in which it cuts the curve is called the principal vertex. 3 3 N Thus: draw N,P,W,, N,P,W2, NI N,P,W,, KASX, through the points N, P,, P2, P3, S, perpendicular to the directrix; each of these lines is a diameter; P,, P,, P., A, are the vertices of these diameters; ASX is the axis of the parabola, A the principal vertex. N3 n A S W 3 6. A straight line which meets the curve in any point, but which, when produced both ways, does not cut it, is called a tangent to the curve at that point. 7. A straight line drawn from any point in the curve, parallel to the tangent at the vertex of any diameter, and terminated both ways by the curve, is called an ordinate to that diameter. 8. The ordinate which passes through the focus, is called the parameter of that diameter. 9. The part of a diameter intercepted between its vertex and the point in which it is intersected by one of its own ordinates, is called the abscissa of the diameter. 10. The part of a diameter intercepted between one of its own ordinates and its intersection with a tangent, at the extremity of the ordinate, is called the sub-tangent of the diameter. Thus: let TPt be the tangent at P, the vertex of the diameter PW. From any point Q in the curve, draw Qq parallel to Tt and cutting Z PW in v. Through S draw RST parallel to Tt. Let QZ, a tangent at Q, cut WP, produced in Z. T Then Qq is an ordinate to the diameter PW; Rr is the parameter of PW. P W Pu is the abscissa of PW, corresponding to the point Q. Z is the sub-tangent of PW, corresponding to the point Q. 11. A stright line drawn from any point in the curve, perpendicular to the axis, and terminated both ways by the curve, is called an ordinate to the axis. 12. The ordinate to the axis which passes through the focus is called the principal parameter, or latus rectum of the parabola. 13. The part of the axis intercepted between its vertex and the point in which it is intersected by one of its ordinates, is called the abscissa of the axis. 14. The part of the axis intercepted between one of its own P ordinates, and its intersection with a tangent at the extremity of the ordinate, is called the sub-tangent of the axis. Thus from any point P in the curve draw Pp perpendicular to AX and cutting AX in M. Through S draw LSI perpendicular to AX. Let PT, a tangent at P, cut XA Tproduced in T. Then, Pp is an ordinate to the axis; LI is the latus rectum of the curve. AM is the abscissa of the axis corresponding to the point P. A S MX MT is the subtangent of the axis corresponding to the point P. It will be proved in Prop. III, that the tangent at the principal vertix is perpendicular to the axis; hence, the four last definitions are in reality included in the four which immediately precede them. Cor. It is manifest from Def. 1, that the parts of the curve on each side of the axis are similar and equal, and that every ordinate Pp is bisected by the axis. 15. If a tangent be drawn at any point, and a straight line be drawn from the point of contact perpendicular to it, and terminated by the curve, that straight line is called a normal. 16. The part of the axis intercepted between the intersections of the normal and the ordinate, is called the sub-normal. Thus: Let TP be a tangent at any point P. From P draw PG perpendicular to the tangent, and PM perpendicular to the axis. Then PG is the normal corresponding to the point P; MG is the sub-normal corresponding to the point P. T S P PROPOSITION I. THEOREM. The distance of the focus from any point in the curve, is equal to the sum of the abscissa of the axis corresponding to that point, and the distance from the focus to the vertex. The latus rectum is equal to four times the distance from the focus to the vertex. To draw a tangent to the parabola at any point. Let P be the given point. Join S, P; draw PN perpendicular to the directrix. n Bisect the angle SPN by the straight N line Tt. Tt is a tangent at the point P. For if Tt be not a tangent, let Tt cut the curve in some other point p. Join S, p; draw pn perpendicular to the directrix; join S, N. TA W Since SP=PN, PO common to the triangles SPO, NPO, and angle SPO=angle NPO by construction, .. SO=NŎ, and angle SOP-angle NOP. Again, since SO-NO, Op common to the triangles SOp, NOP, and angle SOp=angle NOp. ... Sp=Np. But since p is a point in the curve, and pn is drawn perpendicular to the directrix, Sp=pn That is, the hypothenuse of a right-angled triangle equal to one of the sides, which is impossible, .. p is not a point in the curve; and in the same manner it may be proved that no point in the straight line Tt can be in the curve, except P. .. Tt is a tangent to the curve at P. Cor. 1. A tangent at the vertex A, is a perpendicular to the axis. Cor. 2. SP=ST, For, since NW is parallel to TX .. angle STP= angle NPT = angle SPT by construction, SP=ST Cor. 3. Let Qq be an ordinate to the diameter PW, cutting SP in x. Then, Px=Pv For, since Qq is parallel to Tt .. angle Pxv=angle xPT tion, N = NPT by construc = Pux interior oppo- T site angle, .. Px=Pv Cor. 4. Draw the normal PG, (see diagram Prop. V.) For since angle GPT is a right angle, angle GPT=PGT+PTG=PGT+SPT Take away the common angle SPT and there remains angle SPG=angle SGP SP=SG. 17* |