
Conic
Sections 


Parabola
and Line

Common points of a line and a parabola 
Condition for a line to be the tangent to the parabola
 tangency condition

The equation of the tangent and the normal at the point on the parabola

Properties of the parabola






Common points of a line and a parabola 
Common points of a line and a parabola we determine by solving their equations as the system of two
equations in two unknowns, 
(1)
y = mx
+ c 
(2)
y^{2} =
2px
(1)
=> (2)
m^{2}x^{2}
+ 2(mc

p)x + c^{2}
=
0, 
therefore,
the coordinates of intersections of a line and a parabola 


Condition for a line to be the tangent to the parabola
 tangency condition

In the formulas for calculating coordinates of intersections there is the expression under the square root
whose value determines three possible cases regarding mutual position of a line and a parabola, 
so for
p >
0 and, 
p

2mc >
0
 the line intersects the parabola at two points S_{1}(x_{1}, y_{1}) and S_{2}(x_{2}, y_{2}), 
p

2mc
= 0  the line is the
tangent of the parabola and have one point of contact D((p

mc)/m^{2},
p/m)) 
or by substituting p
= 2mc, the
tangency point D(c/m,
2c), 
p

2mc
< 0  the
line and the parabola do not intersect. 
If we write the above conditions as 

then 

these
three cases can be explained graphically as
the relation between parameters
m
and c
of the line and the position of the focus F(p/2,
0).

At the yintercept
N
of the line drawn is a
perpendicular which intersects the xaxis at
M,
then

ON
=  c  and OM
=  c · tan
a

=  m · c .

Therefore, when the point
M
is located;

 to the left of
F, i.e.,
if m
· c
< p/2

the line intersects the parabola at
S_{1}and S_{2},

 to the right
of F, i.e.,
if m
· c >
p/2

the line and the parabola do not intersect,

 at the focus F
or if m
· c
= p/2

the line is the tangent of the parabola.





The equation of the tangent and the normal at the point on the parabola

In the equation of the line
y

y_{1}
= m(
x

x_{1})
through the given point we express the slope m
by the given

ordinate of the tangency point,




and since the coordinate of the tangency point must
satisfy the equation of the parabola, then


obtained
is 
y_{1}y
= p(x
+ x_{1}) 
the
equation of


the tangent at the point
P(x_{1}, y_{1})
on the parabola.

Since


the above equation


can be written
using coordinates of the tangency point





As the slope of the normal 

then
the equation of the normal at P(x_{1}, y_{1}),




or 




Properties of the parabola

Using equations of the tangent and normal expressed by coordinates of the tangency points and
the figure above;

a) yintercept
c_{t}
of the tangent equals half of the ordinates of the tangency point, c_{t}
= y_{1}/2. 
b) the projection
AB
of the segment BP_{1}
of the tangent to the xaxis, i.e., to the axis of the parabola, is
equal to twice the abscissa of the tangency
point, so 
AB
= S_{t}
= 2x_{1 } the line segment
AB
is called the subtangent. 
c) the projection
AC
of the segment CP_{1}
of the normal to the xaxis is equal to the parameter
p, i.e., 
AC
= S_{n}
= p _{ } the line segment
AC
is called the subnormal. 
As points B
and C
are xintercepts of the tangent and the normal their abscissas we determine by solving
corresponding equations for y =
0, so 
put y =
0 into equation of the tangent, 


put
y =
0 into equation of the normal, 



Thus, the focus
F(p/2,
0) bisects the line segment
BC
whose endpoints are xintercepts of the tangent and
the normal, as shows the figure above. 

The tangent at any point on the parabola bisects the angle
j between focal distance and the perpendicular
to the directrix and is equally inclined to the focal distance and the axis of the parabola. 
The normal at the tangency
point bisect the supplementary angle of the angle
j.

Since,
DP_{1}
= FP_{1}
= r
= x_{1}
+ p/2,

and
BF
= FC
= x_{1}
+ p/2
= r, and DP_{1}
 BC

then, following triangles are
congruent,

DBFD
@
DFCP_{1}
@
DFP_{1}D

so, the quadrangle
BFP_{1}D
is the rhombus and its
diagonal BP_{1}
bisects the angle j.

This property is known as the reflective property of the parabola.

A light rays coming in parallel to the axis of a
parabolic mirror (telescope), are reflected so that they all
pass through the focus. Similarly, rays originating at the focus (headlight) will be reflected parallel to the axis.





Tangents drawn at the endpoints of a focal chord intersect at right angles on the
directrix. 
a)
Solving the system of equations of tangents,


x
is the abscissa of the intersection S
of tangents.

The slope of the focal chord line through
tangency points,


therefore,
x = 
p/2 is the abscissa
of the intersection S(
p/2, y) and the equation of the directrix
d.

b)
The tangent to the parabola which passes through intersection
S, which
lies on the directrix, must satisfy tangency condition.





Thus, satisfied is condition for
perpendicularity, m_{1
}
· m_{2}
=

1 =>
j
=
90°. 








Intermediate
algebra contents 



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