PARABOLA

In Geometry, one of the conic sections. formed by the intersection of the cone with a plane parallel to one of its sides. Considered as a plane curve, the parabola may be defined as follows: A point F, and a straight line B B', being given by position in a pkme, let another point D be supposed to move in such a manner that its distance DF from the given point a j i i is always equal to its distance D H from the given straight line; T. A. the point D will trace out the 1 It n o parabola. the The directrix given line of theB B'parabola;is called the given point F is the focus; 1 the straight line F C, drawn through F perpendicular to the directrix, is the axis; any Straight line parallel to C F is a diameter; the point in which the diameter meets the curve is the vertex of the diameter; and a straight line, quadruple the distance between the vertex of any diameter and the directrix, is called the lat us rectum or parameter of that diameter. From the preceding definition of the curve its algebraic equation is easily found. Let A be the origin of the rectangular co-ordinates, A K — x, K!) = ; then F K = — r cos. 6, and consequently CK = 2i — r cos. . But by the definition of the curve, CK = FD = r; therefore r = 2i — r cos. d>,rwhence r = r—1 -J-; cos.■— a)-. This is the polar equation of the parabola. Let T t be a tangent to the parabola through D; it is a property of the curve that the angle H D T, and consequently the vertical angle A 1) t, is equal to F D T. A ray of light, therefore, falling on the curve In the direction A D, and being reflected by it, would pa through the point F; and as this takes place with regard to every ray parallel to the axis, it follows that the concave surface formed by the revolution of a parabola about its axis is that by wliich all the parallel rays of light are collected into a single point. Hence the point F is called the focus, or burning Another remarkable property of the parabola is the following: Let P A be a tangent to the curve at P, and from point.A and B, points in the tangent, let A C and B D be drawn parallel to the diameter P M N; then A C: B D:: P A2: P B*. For, making C M and D N parallel to P B, and as-, suming p be the parameter of the diameter P N, we have, from a property already mentioned, M C- = p- PM, and N D2 =? P N; therefore M C*: N B-2:: P M: P N, or P A* Pit-:. A C: B D. Hence the parabola is the curve described by a projectile in a vacuum; for a body projected from P in the direction P A. and not resisted, would pa over spices in that direction proportional to the times; and", in consequence of the accelerating force of gravity, it falls through spaces A C and B D in the perpendicular direction proportional to the squares of the times, or proportional to the squares of P A and P B. See Gunnery. Projectile. The parabola is remarkable, as being the first curve of which the indefinite quadrature was found. Let P N be a diameter, and P B a tangent at its vertex, and B II and D N respectively parallel to P N and P 1!; it was demonstrated by Arch i me des that the area contained by P N, N D, and the parabolic arc P C D, is equal to two thirds of the paralelogramPB D N. The curve which has now been described is called the conical or Jlpollonian parabola; but the term parabola la [s. 902]