The Parabola

• Definition and equation
• Parametric equations of the parabola
• Tangent line in a point D of a parabola
• Powerfull properties
• Tangent line with a given slope
• Tangent lines from a given point

Definition and equation

Take a line d and a point F not on d.
The locus of all points D such that |D,d| = |D,F| is a parabola.
To obtain an equation, we choose the x-axis and y-axis as in the figure below.

We give F coordinates (p/2,0).
Then we have d with equation x = - p/2.

 
 D(x,y) is on the parabola
 
 <=>
 
 |D,d| = |D,F|
 
 <=>
 
 |D,d|2 = |D,F|2
 
 <=>
 
 p 2 p 2
 (x + -) = (x - -) + y2
 2  2
 <=>
 
 ...
 
 <=>
 
 y2 = 2p x

The point F is called the focus and the line d is the directrix.

Parametric equations of the parabola

Take in a plane two lines a and b with resp. equations

 
 x = 2 p t2 (1)
 
 y = 2 p t (2)

The real numer t is the parameter.
We know, from the theory of 'Elimination of parameters', that the intersection points of the two associated lines constitute a curve. To obtain the equation of that curve, we eliminate the parameter t from the two equations. This means that we search for the condition such that (1) and (2) has a solution for t.
From (2) we have t = y / (2p).
So, this t-value is a solution of (1) if and only if

 
  y 2
 x = 2 p (---) <=> y2 = 2 p x
 2 p
 

Hence, the two associated lines constitute a curve and that curve is the parabola.
We say that (1) and (2) are parametric equations of the parabola. The point

 
D( 2 p t2 , 2 p t)

is on the parabola for each t-value and with each point of the parabola corresponds a t-value.

Tangent line in a point D of a parabola

Take the parabola

 
 y2 = 2p x

To obtain the slope of the tangent line we differentiate implicitly.

 
 2 y y' = 2 p
<=>
 y' = p/y

Say D(x_o,y_o) is a fixed point of the parabola.
The slope of the tangent line in point D is

 
 p
 ---
 y0

The equation of the tangent line is

 
 p
 y - y0 = -- (x - x0)
 y0
<=>
 y0 y - y02 = p x - p x0
 
 Since y02 = 2p x0
 
<=>
 y0 y - 2 p x0 = p x - p x0
<=>
 y y0 = p (x + x0)

The last equation is the tangent line in point D(x₀,y₀) of a parabola.
It is easy to show that this line meets the x-axis at the point s(-x₀,0).
From this it is easy to construct the tangent line in a given point D. (see figure)

Powerfull properties

From this we deduce many properties.
|C,D| = |D,F| = |E,F| = x₀ + p/2 and so CDEF is a rhomb.
Hence the tangent line bisects the angle CDF.
Point C is the mirror image of F with respect to the tangent line.
So, the mirror image of F with respect to a variable tangent line is the directrix.
Additionally, the orthogonal projection of F on a variable tangent line is the tangent line through the vertex of the parabola.
The line through point D and orthogonal with the tangent line is called the normal at point D.
The normal through D is also a bisecting line of CD and DF.

Tangent line with a given slope

Take a line t with a given slope m. The equation is y = m x + q.
The intersection points with the parabola are the solutions of the system

 
 y2 = 2 p x
 
 y = m x + q
 
Substitution gives
 (m x + q)2 = 2 p x
<=>
 m2 x2 + 2 (m q - p) x + q2 = 0
 
The line t is a tangent line if and only if the roots of the last
equation are equal. Therefore the discriminant has to be zero.
 
 4 (m q - p)2 - 4 m2 q = 0
<=>
 4 p (p - 2 m q) = 0
<=>
 p
 q = ---
 2 m
The tangent line with a given slope m is
 
 p
 y = m x + ---
 2 m

Tangent lines from a given point

Take a fixed point P(x₀,y₀) .
We'll calculate the tangent lines from P to the parabola .
A line t with variable slope through P is

 
 y - y0 = m(x - x0)

The intersection points with the parabola are the solutions of the system

 
 y2  = 2 p x
 
 y - y0 = m(x - x0)
 
We substitute x from the first equation into the second one.
 y2
 y - y0 = m(--- - x0)
 2 p
<=>
 - m y2 + 2 p y - 2 p y0 + 2 p m x0 = 0

The line t is a tangent line if and only if the roots of the last equation (in y) are equal. Therefore the discriminant has to be zero.

 
 4 p2 + 4 m (2 p m x0 - 2 p y0) = 0
<=>
 2 x0 m2 - 2 y0 m + p = 0

The roots of this equation are the slopes of the two tangent lines.

 
 
 y0 + sqrt(y02 - 2 p x0)
 m1 = ------------------------------
 2 x0
 
 y0 - sqrt(y02 - 2 p x0)
 m2 = ----------------------------
  2 x0

The equations of the tangent lines are

 
 y2
 y - y0= m1(----- - x0) ;
 p
 
 y2
 y - y0= m2(----- - x0)
 p
 

The two lines are orthogonal if and only if m₁ . m₂ = -1

 
 p
<=> ---- = -1
 2 x0
 
 
 p
<=> x0 = ----
 2

From this we see that if point P is on the directrix, the tangent lines are orthogonal.