
Conic
Sections 


Parabola
and Line

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

Polar 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,.e.,





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

1 => j
=
90°. 

Polar of the parabola

The polar p
of a point A(x_{0}, y_{0}), exterior to the parabola
y^{2} = 2px, is the secant through the contact
points of the
tangents drawn from the point A
to the parabola. 
The tangency points
D_{1}(x_{1}, y_{1})
and D_{2}(x_{2}, y_{2})
and the point A
satisfy the equations of tangents,

t_{1}
::_{ }y_{1}y_{0}
= p(x_{0}
+ x_{1})
and t_{2
} ::
y_{2}y_{0}
= p(x_{0}
+ x_{2}).

Subtracting
t_{2}

t_{1},

y_{0}(y_{2}

y_{1})
= p[(x_{0}
+ x_{2})
 (x_{0}
+ x_{1})]


obtained is the slope of the polar. By plugging the slope into equation of the line through the given point


or
y_{0}y
= y_{1}y_{0}
+ px 
px_{1} 

since y_{1}y_{0}
= p(x_{0}
+ x_{1}),

then

y_{0}y
= p(x
+ x_{0}) 
the
equation of the
polar.













Conic
sections contents 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 