..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(xo,yo) 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.
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