Parabola

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Figure 1: The Parabola
Focus at point
Vertex at origin
Directrix is line
By definition

In Figure 1


In Cartesian geometry in two dimensions the is the locus of a point that moves so that it is always equidistant from a fixed point and a fixed line. The fixed point is called the and the fixed line is called the . Distance from to is non-zero.

See Figure 1.

The focus is point and the directrix is line The , point , is half-way between focus and directrix. A is the segment of a line joining any two distinct points of the parabola. The line segment is a chord. Because chord passes through the focus , it is called a

The focal chord parallel to the directrix is called the

The line through the focus and perpendicular to the directrix is the , sometimes called .


Let an arbitrary point on the curve be .


By definition, . This expression expanded gives:

.


If the equation of the curve is expressed as: , then where the has coordinates and is the distance form vertex to focus.


If the directrix is parallel to the axis, then the parabola is the same as the familiar quadratic function.


The general parabola allows for a directrix anywhere with any orientation.


The General Parabola[edit | edit source]


Let the directrix be where at least one of is non-zero.

Let the focus be .

Let be any point on the curve.


Distance from point to focus = .

Distance from point to directrix ()

= where .


By definition these two lengths are equal:

.

.

Square both sides:

.

.


Expand and the result is:

.


has the form of the equation of the conic section where



because this curve is a parabola.


An Example[edit | edit source]

Figure 2: The Parabola
Green line is
Blue line is
Focus at point
Vertex at point
Shape of curve is:

See Figure 2.



The equation of the parabola is derived as follows:



The equation of the parabola in Figure 2 is:


Equation of directrix in normal form:

Distance from to

Distance from vertex to focus .

Therefore, curve has shape of where .

Caution: An interesting situation occurs if the focus is on the directrix. Consider the directrix:

and the focus which is on the directrix.


In this case the "parabola" has equation: .

This seems to be the equation of a parabola because , but look closely.

.

The result is a line through the focus and normal to the directrix.


If you solve for using the algebraic solutions, you will produce the values as above.

However, the distance between focus and directrix =

where distance

Reverse-Engineering the Parabola[edit | edit source]

Figure 3: Parabola with 2 tangents parallel to axes.
Tangent .
Tangent .
Point on directrix, oblique, thin, black line.
Line perpendicular to focal chord
Focus at .


Given a parabola in form the aim is to produce the directrix and the focus.


We will solve the example shown in Figure 3: ,


where:


.


Method 1. By analytical geometry[edit | edit source]

Find two tangents that intersect at a right angle. The simplest to find are those that are parallel to the axes.

Put the equation of the parabola in the form of a quadratic in .

At the tangent there is exactly one value of . Therefore the discriminant must be .

In the general parabola therefore .

In this example .

Point has coordinates .

The line is tangent to the curve at and has equation: .

Put the equation of the parabola in the form of a quadratic in :

.

By using calculations similar to the above, and .

Point has coordinates .

The line is tangent to the curve at and has equation: .

Point at the intersection of the two tangents has coordinates , and point is on the directrix, the equation of which is: .

Put known values into the equation of the directrix: .

Therefore and the equation of the directrix is: .

The coordinates of points are known. Therefore chord is defined as: .

Draw the line perpendicular to . The line is defined as: .

Point at the intersection of lines is the focus with coordinates .

Method 2. By algebra[edit | edit source]


After rearranging the above values, there are three equations to be solved for three unknowns: :


The solutions are:




If is the parabola becomes the quadratic: and:

The directrix has equation: .


If is , then:

The directrix has equation: .

If is the parabola becomes the quadratic: and:

The directrix has equation: .

Graph of quadratic function
showing :
* X and Y intercepts in red,
* vertex and axis of symmetry in blue,
* focus and directrix in pink.


If is , then:

The directrix has equation: .


These values agree with the corresponding values in the graph of to the right.

Slope of the Parabola[edit | edit source]


Consider parabola and line

Let point be any point on the line. Therefore

Let the line intersect the parabola. Substitute the above value of into the equation of the parabola and expand:



We want the line to be tangent to the curve. Therefore must have exactly one value and the discriminant is

Discriminant =


The above discriminant is a quadratic in :



where:


If the point is on the curve, then the line touches the curve at and:



because for a parabola.


When slope is displayed in this format, we see that slope is vertical if


The line is tangent to the curve.

Let the equation of a line be: in which case

By using calculations similar to the above it can be shown that:


.


therefore:



When slope is displayed in this format, we see that slope is zero if .


The line is tangent to the curve.

This examination of the parabola has produced two expressions for slope of the parabola:


.


where the point is any arbitrary point on the curve.


Therefore . This formula for contains both tangents parallel to the axes and is derived without calculus.

The formula from calculus below is simpler and unambiguous concerning sign.


The slope of the parabola is where or:


The slope of the parabola is vertical where or:

Caution:

If the curve is , the slope can never be vertical.

If the curve is , the slope can never be .

If and , the equation of the parabola becomes: and:


Point at given slope[edit | edit source]

Given parabola defined by and slope where at least one of is non-zero, calculate point at which the slope is


Let


Then


Let


Then

Substitute in the equation of the parabola and

As shown below, with a little manipulation of the data, the same formula can be used to calculate

Equation above is the equation of a straight line with slope

Substitute for and the slope of becomes

Equation is that of a line parallel to the axis of symmetry of the parabola.

It is possible for both both to equal in which case the calculation of above fails as an attempt to divide by See caution under "Slope of the Parabola" above.

Implementation[edit | edit source]

# python code
def pointAtGivenSlope(parabola, tangent) :
    s,t = tangent
    if s == t == 0 :
        print ('pointAtGivenSlope(): both s,t can not be 0.')
      	return None
    def calculate_y (parabola, tangent)  :
        A,B,C,D,E,F = parabola
        s,t = tangent
      	G = B*s + 2*A*t	; H = 2*C*s + B*t ; I =	E*s + D*t
      	return -(A*I*I - D*G*I + F*G*G) / (2*A*H*I - B*G*I - D*G*H + E*G*G)
    y = calculate_y (parabola, tangent)
    A,B,C,D,E,F = parabola
    x = calculate_y ((C,B,A,E,D,F), (t,s))
    return x,y

Examples[edit | edit source]

A parabola is defined as

Calculate coordinates of vertex.

At vertex tangent has same slope as directrix.

# python code
parabola = A, B, C, D, E, F = 9, -24, 16, 70, -260, 1025
a = C**.5
b = -B/(2*a)
tangent = -a,b
result = pointAtGivenSlope(parabola, tangent)
print (result)
(0.2, 6.4)

Calculate point at which tangent is vertical.

# python code
parabola = A, B, C, D, E, F = 9, -24, 16, 70, -260, 1025
tangent = 1,0
result = pointAtGivenSlope(parabola, tangent)
print (result)
(-0.25, 7.9375)

Parabola and any chord[edit | edit source]


Figure 4: Parabola and any chord.
Origin at point curve
chord point
2 tangents point

Refer to Figure 4.

The curve has equation:

The chord has equation: .

Point has coordinates

Line is tangent to the curve at

Line is tangent to the curve at

This section shows that point has coordinates



where


Point has coordinates where:

,

,

and slope of tangent


Point has coordinates where:

,

,

and slope of tangent


Points have coordinates


Equation of tangent


Equation of tangent


At point of intersection


Review the coordinates of points .

The line with equation bisects the line segment and also the chord at point .


Any chord parallel to has two tangents that intersect on the line .

Any chord parallel to is bisected by the line .


The coordinate of point



Any chord that passes through the point has two tangents that intersect on the line .


Angle

Using:


If , point is above the is positive and ° °.

is acute and, as increases, increases, approaching °.


If , point is on the ° and the line is the .


If , point is below the is negative and ° °.

is obtuse and, as increases, decreases, approaching °.

Parabola and two tangents[edit | edit source]


Figure 5: Parabola and two tangents.
Origin at point curve
point 2 tangents
chord point

Refer to Figure 5.

The curve has equation:

Point with coordinates is any point on the line

Line is tangent to the curve at point .

Line is tangent to the curve at point .

This section shows that the chord passes through the point


Equation of any line through point


Let this line intersect the curve:


We want this line to be a tangent to the curve, therefore has exactly one value and the discriminant is :



where


slope of tangent .

slope of tangent .



Slope of chord


Intercept of chord on the axis


Angle



If °

In the basic parabola where the has coordinates .

or

If then and are the same point, the chord passes through the and the line is the .


Area enclosed between parabola and chord[edit | edit source]

Introduction[edit | edit source]

Figure 6: The Parabola:
Chord , parallel tangent and area

Chord , parallel tangent and area .

See Figure 6. The curve is: . Integral is: .


Area under curve

Area under curve area area

Area between chord and curve .


Consider the chord . Call this the with value . The tangent through the origin is parallel to , and the perpendicular distance between is . Call this distance the with value .

In this case the area enclosed between chord and curve the same as that calculated earlier.


Consider the chord and curve By inspection the area between chord and curve


Chord has equation in normal form.

The line is parallel to and touches the curve at

Distance from to chord

Length of

Area between chord and curve the same as that calculated earlier.


These observations suggest that the area enclosed between chord and curve

Proof[edit | edit source]

Figure 7: The Parabola:
Origin at point
Points
Points
Chord , parallel tangent and area , the area enclosed between chord and curve.
Length Length

We prove this identity for the general case. See Figure 7.

Slope of chord

Find equation of chord

Equation of chord


Find equation of tangent

We choose a value of that gives exactly one value.

Therefore discriminant

Equation of tangent in normal form:


Equation of chord in normal form:

Therefore distance between chord and tangent

where


Length of chord


Area under chord


Area under curve


Area between chord and curve


The aim is to prove that:

or

or

or

or


where:


Therefore and area enclosed between curve and chord =

where is the length of the chord, and is the perpendicular distance between chord and tangent parallel to chord.

A worked example[edit | edit source]

Consider parabola:


and chord:


The aim is to calculate area between chord and curve.


Calculate the points at which chord and parabola intersect:


Method 1. By chord and parallel tangent[edit | edit source]

Figure 8: The Parabola:
Chord
Length
Parallel tangent
Area between chord and curve

Length of chord



Equation of chord in normal form:


Equation of parallel tangent in normal form:


where


and


and



Equation of chord in normal form:


Equation of parallel tangent in normal form:


Distance between chord and parallel tangent


Area enclosed between chord and curve

Method 2. By identifying the basic parabola.[edit | edit source]

Figure 9: The Parabola:
Chord
is line is line
is point
Line Line

See Figure 9. Calculate directrix, focus and axis.


Focus is distance from directrix. Curve has the shape of where .


Axis of symmetry has equation:


Distance from point to axis of symmetry


Distance from point to axis of symmetry



Area between chord and curve


or:


where


Calculate as above and area enclosed between chord and curve .