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.

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

In Figure 1


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

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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

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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

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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

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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

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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

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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

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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

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# 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
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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

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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

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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

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Introduction

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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

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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

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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

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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.

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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  .