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Parabolas

Definitions

General Equation of a Parabola

Standard Forms of a Parabola

Reduction to Standard Form

Parabola from Focus and Directrix

Parametric Equations

Explorations

In the branch of mathematics known as celestial mechanics it is shown that an object, such as a comet, that falls toward the sun "from infinity" would, if not deflected by the gravitational attraction of other bodies, travel in a path whose shape is a parabola with the sun at its focus. Projectiles in a vacuum on the surface of the earth travel in paths which are nearly parabolic, and projectiles in the air approximate this path with greater or less precision according to their speed, shape and weight. Humans also take advantage of the focusing properties of a parabolic shape in the design of such artifacts as headlights, searchlights and various listening and broadcasting devices. This chapter develops the underlying mathematics of a parabola.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Definitions [Top]

A parabola is the locus of a point that moves so that the ratio of its distance from a fixed point and from a fixed line is one. The fixed point, F, is called the focus and the fixed line, D, the directrix. By definition, the distance from any point P on the parabola to the focus is equal to its distance to the directrix. The ratio PF/PD is called the eccentricity e and e=1. The line FD through the focus perpendicular to the directrix is called the axis of the parabola. The midpoint V of the segment FD, obviously a point on the locus, is called the vertex of the parabola. The focal chord perpendicular to the axis is called the latus rectum.

Definition of a parabola.

General Equation of a Parabola [Top]

We choose any point as the focus and any line , where , as the directrix. The normalized form of the line is used to simplify the derivation. With reference to these defining elements the equation of the parabola becomes

which can be written as

This equation is of the form , an equation of the second degree. One characteristic of the equation is that the second-degree terms in x and y form a perfect square, so the equation may also be written

Moreover, it can be verified that . Therefore a necessary condition that the equation represent a parabola is that . The general equation of a parabola reveals that if the directrix line is parallel to one of the coordinate axes then B=0 since either or will be zero. The equation of a parabola in this position will have no xy term.

Standard Forms of a Parabola [Top]

The definition of a parabola makes the shape of the curve depend only upon the distance from the focus to the directrix and not essentially upon the coordinate system. The general equation is complicated because of the choice of a general point and a general line. By an appropriate choice of axes this equation can be simplified; but it will then represent only parabolas in special positions. For example, if axes are chosen so that the focus has coordinates (f,0) and the directrix the equation x=-f, then the locus definition yields

which reduces to . This is one of the standard forms of the equation of a parabola. It has a vertex V(0,0). If f is positive the parabola opens to the right; if f is negative it opens to the left. The distance f is called the focal length of the parabola and is the distance between the focus and the vertex of the parabola.

Generalizing the location of the vertex point to V(h,k) gives a parabola whose equation is

This equation is the standard form used in Descarta2D as illustrated in the following example.

Example: Plot the parabola whose vertex is (2,1), focal length is 1/2, and opens to the right.

Solution: Parabola2D[{h,k},f,θ] is the standard representation of a parabola in Descarta2D where the coordinates of the vertex are (h,k), the focal length is f and the rotation angle (in radians) about the vertex is θ.

Sketch2D[{Parabola2D[{2,1},1/2,0]}]

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The axis of the parabola may also be parallel to the y-axis in which case the equation is

Descarta2D does not directly use this form of the parabola, but instead simply rotates the form by the appropriate angle.

Example: Plot the parabola whose vertex is (1,-1), focal length is 1/3, and opens upward.

Solution: Use the same command as in the previous example with a rotation angle of π/2 radians.

Sketch2D[{Parabola2D[{1,-1},1/3,Pi/2]}]

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Example: Plot the four parabolas whose vertices are (1,1), (-1,1), (-1,-1) and (1,-1), focal length 1/3, and axes aligned with the lines x-y=0 and x+y=0.

Solution: The Descarta2D command Parabola2D[{h,k},f,θ] returns the desired parabolas using the Angle2D[line] command to find the required values for θ.

axis1=Line2D[1,-1,0]; axis2=Line2D[1,1,0];

theta={Angle2D[axis1],Angle2D[axis2],

Angle2D[axis1]+Pi,Angle2D[axis2]+Pi};

pts={{1,1},{-1,1},{-1,-1},{1,-1}};

Sketch2D[{axis1,axis2,

Map[Parabola2D[pts[[#]],1/3,theta[[#]]]&,

Range[1,4]]}]

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Reduction to Standard Form [Top]

The most general equation of a parabola with no xy term present (and hence one whose axis is parallel to one of the coordinate axes) is one of the two forms

(1) | axis parallel to the y-axis; | |

(2) | axis parallel to the x-axis. |

In either case it is easy to reduce this general equation to the corresponding standard form by the process of completing the square.

Example: Reduce to the equation of a parabola in standard form.

Solution: The Descarta2D function Loci2D[quadratic] returns a list of geometric objects represented by a quadratic equation.

crv1=Loci2D[Quadratic2D[1,0,0,4,4,-8]]

The equation in standard form is , rotated (3π/2 radians) about the point (-2,3).

Sketch2D[{crv1}]

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The following example shows how Descarta2D may be used to find the various geometric objects associated with a parabola.

Example: Find the focus, directrix, vertex, axis and eccentricity of the parabola represented by the equation . Plot the geometric objects.

Solution: The function Foci2D[parabola] returns a list of one point which is the focus of the parabola; Directrices2D[parabola] returns a list of one line which is the directrix of the parabola; Line2D[parabola] returns the axis line of the parabola; and the function Eccentricity2D[parabola] returns the eccentricity of a parabola (always 1).

p1=First[Loci2D[Quadratic2D[1,0,0,-2,-8,-15]]]

{Eccentricity2D[p1],

geom=Map[(#[p1])&,

{Foci2D,Directrices2D,Vertices2D,Line2D}]} //Flatten

Sketch2D[{p1,geom}]

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Parabola from Focus and Directrix [Top]

A parabola may be defined in terms of a focus point and a directrix line given by . Given these two defining elements the parabola's parameters (vertex point V(h,k), focal length f and angle of rotation θ) can be determined. Let

be the signed distance from the focus F to the directrix L, D=|d|, and F' be the projection of F on L. From a previous chapter the coordinates of F' are given by , where and . The vertex point V is obviously the midpoint of the line segment FF' and has coordinates . The focal length f is half of D, f=D/2. The rotation angle θ is the angle of the line FF'.

Example: Determine the parabola in standard form defined by the focus point F(1,1) and the directrix line x+y=0.

Solution: The Descarta2D function Parabola2D[point,line] returns the parabola defined by a focus point and a directrix line.

p1=Parabola2D[Point2D[{1,1}],Line2D[1,1,0]]

{Foci2D[p1],Directrices2D[p1]} //Simplify

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Parametric Equations [Top]

The standard form of a parabola used in Descarta2D has the equation

where (h,k) is the vertex of the parabola and f is the focal length. The axis of this parabola is parallel to the x-axis and the parabola opens to the right (when f>0). Parabolas in other orientations are obtained by applying a rotation, θ, to the standard parabola. Since only the y term is quadratic, it is easy to find one set of parametric equations for a parabola. Let y=k+2ft be one of the equations; then, solving for t yields

Substituting this into the equation of the parabola and solving for x yields the two parametric equations

x | = | |

y | = | k+2ft. |

The parameter value t=0 produces the vertex point (h,k). Increasing values of t produce points above and to the right of the vertex. Negative values of t produce points that correspond to positive t values reflected in the axis of the parabola. Parameter values t=±1 produce the end points of the focal chord of the parabola.

Example: Generate seven points on the parabola at equally spaced parameter values. Plot the curve using a curve length of 20. Generate a second plot of the points on the reflected branch of the parabola.

Solution: The Descarta2D command Parabola2D[{h,k},f,θ][t] returns the coordinates at parameter t on the parabola. The option CurveLength2D->n sets the approximate length of unbounded curves plotted by Descarta2D.

p1=Parabola2D[{1,-1},1/2,0];

pts1=Map[Point2D[p1[#/2]]&,Range[0,6]];

pts2=Map[Point2D[p1[#/2]]&,Range[-6,0]];

{Sketch2D[{p1,pts1},CurveLength2D->20],Sketch2D[{p1,pts2},CurveLength2D->20]}

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Mathematica Hint: Using the CurveLength2D option as part of the Sketch2D command sets the length of all unbounded curves being plotted. If this option is not specified, then a default is used. The initial default set by Descarta2D is 10. To change the default to a new value, n, use the Mathematica command

SetOptions[Sketch2D,CurveLength2D->n].

Explorations [Top]

Length of Parabola Focal Chord. [pbfocchd.html]

Prove that the length of the focal chord of a parabola is 4f, where f is the focal length.

Parabola Through Three Points. [pb3pts.html]

Show that the parabola passing through the points (0,0), (a,b) and (b,a) whose axis is parallel to the x-axis has vertex (h,k) and focal length f given by

Furthermore, show that the quadratic representing the parabola is

Parabolic Arch. [pbarch.html]

Find the equation of the parabolic arch of base b and height h as shown in the figure. Assume that b and h are positive.

Parabola Determinant. [pbdet.html]

Show that the determinant

represents a parabola passing through the points , and .

Parabola Intersection Angles. [pbang.html]

Show that the parabolas and will cut each other at an angle θ given by

Circle Tangent to a Parabola. [pbtancir.html]

Any line through the point (-3a,0) cuts the parabola in the points P and Q. Prove that the circle through P, Q and the focus is tangent to the parabola.

Polar Equation of a Parabola. [polarpb.html]

Show that the polar equation of a parabola opening to the right with vertex at (0,0) is given by

where f is the focal length of the parabola.

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