PARABOLA, a plane curve of the second degree. It may be defined as a section of a right circular cone by a plane parallel to a tangent plane to the cone, or as the locus of a point which moves so that its distances from a fixed point and a fixed line are equal. It is therefore a conic section having its eccentricity equal to unity. The parabola is the curve described by a projectile which moves in a nonresisting medium under the influence of gravity (see MECHANICS). The general relations between the parabola, ellipse and hyperbola are treated in the articles GEOMETRY, ANALYTICAL, and CONIC SECTIONS; and various projective properties are demonstrated in the article GEOMETRY, PROJECTIVE. Here only the specific properties of the parabola will be given.
The form of the curve is shown in fig. 1, where P is a point on the curve equidistant from the fixed line AB, known as the directrix, and the fixed point F known as P the focus. The line CD passing through the focus and perpendicular to the directrix is the axis or principal diameter, and meets the curve in the vertex G. The line FL perpendicular to the axis, and passing through the focus, is the semilatus rectum, the latus rectum being the focal chord parallel fo the directrix. Any line parallel to the axis is a diameter, and the parameter of any diameter is measured by the focal chord drawn FIG. I.
parallel to the tangent at the vertex of the diameter and is equal
B
748
to four times the focal distance of the vertex. To construct the parabola when the focus and directrix are given, draw the axis CD and bisect CF at G, which gives the vertex. Any number of points on the parabola are obtained by taking any point E on the directrix, joining EG and EF and drawing FP so that the angles PFE and DFE are equal. Then EG produced meets FP in a point on the curve. By joining the points so obtained the parabola may be described. A mechanical construction, when the same conditions are given, consists in taking a rigid bar ABC bent at right angles at B (fig. 2),
and fastening a string of length BC to C
and F. Then if a pencil be placed along
/p BC so as to keep the string taut, and the
limb AB be slid along the directrix, the n pencil will trace out the parabola.
F Properties which may be readily de
A fundamental property of the curve is that the line at infinity is a tangent (see GEOMETRY, PROJECTIVE), and it follows that the centre and the second real focus and directrix are at infinity. It also follows that a line halfway between a point and its polar and parallel to the latter touches the parabola, and therefore the lines joining the middle points of the sides of a selfconjugate triangle form a circumscribing triangle, and also that the ninepoint circle of a selfconjugate triangle passes through the focus. The orthocentre of a triangle ,circumscribing a parabola is on the directrix; a deduction from this theorem is that the centre of the circumcircle of a selfconjugate triangle is on the directrix (" Steiner's Theorem ").
In the article GEOMETRY, ANALYTICAL, it is shown that the
general equation of the second degree represents a parabola
when the highest terms form a perfect square.
Analytic This is the analytical expression of the projective
Geometry.
property that the line at infinity is a tangent. The simplest equation to the parabola is that which is referred to its axis and the tangent at the vertex as the axes of coordinates, when it assumes the form y2=4ax where 2a=semilatus rectum; this may be deduced directly from the definition. An equation of similar form is obtained when the axes of coordinates are any diameter and the tangent at the vertex. The equations to the tangent and normal at the point x'y' are yy'= 2a(x+x') and 2a(y—y')+y'(x—x')=o, and may be obtained by general methods (see GEOMETRY, ANALYTICAL, and INFINITESIMAL CALCULUS). More convenient forms in terms of a single parameter are deduced by substituting xi = amt, y' =tam (for on eliminating m between these relations the equation to the parabola is obtained). The tangent then becomes my=x+amt and the normal y=mx+2am—am3. The envelope of this last equation is 27ay2=4(x—2a)3, which shows that the evolute of a parabola is a semicubical parabola (see below Higher Orders). The cartesian equation to a parabola which touches the coordinate axes is V ax+1/ by= 1, and the polar equation when the focus is the pole and the axis the initial line is r cos2O/2=a.
The equation to a parabola in triangular coordinates is generally derived by expressing the condition that the line at infinity is a tangent in the equation to the general conic. For example, in trilinear coordinates, the equation to the general conic circumscribing the triangle of reference is l0y+mya+naf=o; for this to be a parabola the line as + b/3 + cy=o must be a tangent. Expressing this condition we obtain
V la* 1/ml Vnc=o as the relation which must hold between the coefficients of the above equation and the sides of the triangle of reference for the equation to represent a parabola. Similarly, the conditions for the inscribed conic S/la+V m#+V ny = o to be a parabola is lbc+mca+nab=o, and the conic for which the triangle of reference is selfconjugate lag+ml2+ny2=c is a2mn+b2nl+c2lm=o. The various forms in areal coordinates may be derived from the above by substituting Xa for 1, ub for m and vc for n, or directly by expressing the condition for tangency of the line x+y+z=o to the conic expressed in areal coordinates. In tangential (p, q, r) coordinates the inscribed and circumscribed conics take the forms Xqr+µrp+vpq=o and V Ap+ A/,uq±V ye = o; these are parabolas when X +µ+v = o and V X= V,u* V v= o respectively.
The length of a parabolic arc can be obtained by the methods of the infinitesimal calculus; the curve is directly quadrable, the area of any portion between two ordinates being two thirds of the circumscribing parallelogram. The pedal equation with the focus as origin is p2=ar; the first positive pedal for the vertex is the cissoid (q.v.) and for the focus the directrix. (See INFINITESIMAL CALCULUS.)
REFERENcEs.—Geometrical constructions of the parabola are to be found in T. H. Eagles' Plane Curves (1885). See the bibliography to the articles CONIC SECTIONS; GEOMETRY, ANALYTICAL; and GEOMETRY, PROJECTIVE.
In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a'nxn= ye.+n, thus ax=y2 is the quadratic or Apollonian
parabola; a2x=y3 is the cubic parabola, a3x=y4 is Higher
Orders.
the biquadratic parabola; semi parabolas have the
general equation axn1=yn, thus ax2=y3 is the semicubical parabola and ax3=y4 the semibiquadratic parabola. These curves were investigated by Rene Descartes, Sir Isaac Newton, Colin Maclaurin and others. Here we shall treat only the more important forms.
The cartesian parabola is a cubic curve which is also known as the trident of Newton on account of its threepronged form. Its equation is xy=ax3+bx2+cx+d, and it consists of two legs asymptotic to the axis of y and two parabolic legs (fig. 3). The simplest form is axy=x3—a3, in this case the serpentine position shown in the figure degenerates into a point of inflexion. Descartes used the curve to solve sextic equations by determining its intersections with a circle; mechanical constructions were given by Descartes (Geometry, lib. 3) and Maclaurin (Organica geometrica).
The cubic parabola (fig. 4) is a cubic curve having the equation y=ax3+bx2+cx+d. It consists of two parabolic branches tending in opposite directions. John Wallis utilized the intersections of this curve with a right line to solve cubic equations, and Edmund Halley solved sextic equations with the aid of a circle.
Diverging parabolas are cubic curves given by the equation y2=ax3+bx2+cx+d. Newton discussed the five forms which arise from the relations of the roots of the cubic equation. When all the
roots are real and unequal the curve consists of a closed oval and a parabolic branch (fig. 5). As the two lesser roots are made more and more equal the oval shrinks in size and ultimately becomes a real conjugate point, and the curve, the equation of which is y2= (x—a)2(x—b) (in which a>b) consists of this point and a belllike branch resembling the righthand member of fig. . If two roots are imaginary the equation is y2 _ (x2+a2) (x —b) and the curve resembles the parabolic branch, as in the preceding case. This is sometimes termed the cam paniform (or bellshaped) parabola. If the two greater roots are equal the equation is y2 (x —a) (x — b)2 (in which a
End of Article: PARABOLA 

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