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PARABOLA

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Originally appearing in Volume V20, Page 749 of the 1911 Encyclopedia Britannica.
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PARABOLA , a See also:

plane See also:curve of the second degree. It may be defined as a See also:section of a right circular See also:cone by a plane parallel to a tangent plane to the cone, or as the See also:locus of a point which moves so that its distances from a fixed point and a fixed See also: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 non-resisting See also:medium under the See also:influence of gravity (see See also:MECHANICS). The See also:general relations between the parabola, See also:ellipse and See also:hyperbola are treated in the articles See also:GEOMETRY, See also:ANALYTICAL, and CONIC SECTIONS; and various projective properties are demonstrated in the See also:article GEOMETRY, PROJECTIVE. Here only the specific properties of the parabola will be given. The See also: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 See also:focus. The line CD passing through the focus and perpendicular to the directrix is the See also:axis or See also:principal See also: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 See also: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 See also: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 See also:

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 See also:mechanical construction, when the same conditions are given, consists in taking a rigid See also:bar See also:ABC See also:bent at right angles at B (fig. 2), and fastening a See also:string of length BC to C and F. Then if a See also:pencil be placed along /p BC so as to keep the string taut, and the See also:limb AB be slid along the directrix, the n pencil will trace out the parabola. F Properties which may be readily de- A fundamental See also: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 See also:half-way between a point and its polar and parallel to the latter touches the parabola, and therefore the lines joining the See also:middle points of the sides of a self-conjugate triangle form a circumscribing triangle, and also that the nine-point circle of a self-conjugate triangle passes through the focus. The orthocentre of a triangle ,circumscribing a parabola is on the directrix; a See also:deduction from this theorem is that the centre of the circumcircle of a self-conjugate triangle is on the directrix (" See also:Steiner's Theorem "). In the article GEOMETRY, ANALYTICAL, it is shown that the general See also:equation of the second degree represents a parabola when the highest terms form a perfect square.

See also:

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 co-ordinates, when it assumes the form y2=4ax where 2a=semilatus rectum; this may be deduced directly from the See also:definition. An equation of similar form is obtained when the axes of co-ordinates 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' =See also: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 semi-cubical parabola (see below Higher Orders). The cartesian equation to a parabola which touches the co-See also:ordinate axes is V ax+1/ by= 1, and the polar equation when the focus is the See also:pole and the axis the initial line is r cos2O/2=a. The equation to a parabola in triangular co-ordinates is generally derived by expressing the See also:condition that the line at infinity is a tangent in the equation to the general conic. For example, in trilinear co-ordinates, 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 co-efficients 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 self-conjugate lag+ml2+ny2=c is a2mn+b2nl+c2lm=o. The various forms in areal co-ordinates 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 co-ordinates. In tangential (p, q, r) co-ordinates 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 See also:

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 See also:positive pedal for the vertex is the See also: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 See also: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 See also:biquadratic parabola; semi parabolas have the general equation axn-1=yn, thus ax2=y3 is the semicubical parabola and ax3=y4 the semibiquadratic parabola. These curves were investigated by Rene See also:Descartes, See also:Sir See also:Isaac See also:Newton, See also:Colin See also: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 See also:

trident of Newton on See also:account of its three-pronged 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 See also:case the See also:serpentine position shown in the figure degenerates into a point of See also:inflexion. Descartes used the curve to solve sextic equations by determining its inter-sections with a circle; mechanical constructions were given by Descartes (Geometry, See also: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. See also:John See also:Wallis utilized the intersections of this curve with a right line to solve cubic equations, and See also:Edmund See also: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 See also:

oval and a parabolic See also:branch (fig. 5). As the two lesser roots are made more and more equal the oval shrinks in See also: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 See also:bell-like branch resembling the right-See also:hand member of fig. . If two roots are imaginary the equation is y2 _ (x2+See also:a2) (x —b) and the curve resembles the parabolic branch, as in the preceding case. This is some-times termed the See also:cam paniform (or bell-shaped) parabola. If the two greater roots are equal the equation is y2-- (x —a) (x — b)2 (in which a<b) and the curve assumes the form shown in fig. 6, and is known as the nodated parabola. Finally, if all the roots are equal, the equation becomes y2=(x—a)3; this curve is the cuspidal or semi-cubical parabola (fig. 7). This curve, which is sometimes termed the Neilian parabola after See also:William Neil (1637-1670), is the evolute of the See also:ordinary parabola, and is especially interesting as being the first curve to be rectified. This was accomplished in 1657 by Neil in See also:England, and in 1659 by Heinrich See also:van Haureat in See also:Holland.

Newton showed that all the five varieties of the diverging parabolas may be exhibited as plane sections of the solid of revolution of the semi-cubical parabola. A plane oblique to the axis and passing below the vertex gives the first variety; if it passes through the vertex, the second form; if above the vertex and oblique or parallel to the axis, the third form; if below the vertex and touching the See also:

surface, the See also:fourth form, and if the plane contains the axis, the fifth form results (see CURVE). The biquadratic parabola has, in its most general form, the equa•• tion y=ax++bx8+cx2+dx+e, and consists of a serpentinous and two parabolic branches (fig. 8). If all the roots of the quartic in x are equal the curve assumes the form shown in fig. 9, the axis of x being a See also:double tangent. If the two middle roots are equal, fig. so results. Other forms which correspond to other relations between the roots can be readily deduced from the most general form.

End of Article: PARABOLA

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