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Conic
Sections |
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Parabola
and Line
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Common points of a line and a parabola |
Condition for a line to be the tangent to the parabola
- tangency condition
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The equation of the tangent and the normal at the point on the parabola
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Properties of the parabola
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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)
y2 =
2px
(1)
=> (2)
m2x2
+ 2(mc
-
p)x + c2
=
0, |
therefore,
the coordinates of intersections of a line and a parabola |
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Condition for a line to be the tangent to the parabola
- tangency condition
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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 S1(x1, y1) and S2(x2, y2), |
p
-
2mc
= 0 - the line is the
tangent of the parabola and have one point of contact D((p
-
mc)/m2,
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 |
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then |
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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).
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At the y-intercept
N
of the line drawn is a
perpendicular which intersects the x-axis at
M,
then
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ON
= | c | and OM
= | c · tan
a
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= | m · c |.
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Therefore, when the point
M
is located;
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- to the left of
F, i.e.,
if m
· c
< p/2
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the line intersects the parabola at
S1and S2,
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- to the right
of F, i.e.,
if m
· c >
p/2
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the line and the parabola do not intersect,
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- at the focus F
or if m
· c
= p/2
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the line is the tangent of the parabola.
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The equation of the tangent and the normal at the point on the parabola
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In the equation of the line
y
-
y1
= m(
x
-
x1)
through the given point we express the slope m
by the given
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ordinate of the tangency point,
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and since the coordinate of the tangency point must
satisfy the equation of the parabola, then
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obtained
is |
y1y
= p(x
+ x1) |
the
equation of
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the tangent at the point
P(x1, y1)
on the parabola.
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Since
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the above equation
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can be written
using coordinates of the tangency point
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As the slope of the normal |
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then
the equation of the normal at P(x1, y1),
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or |
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Properties of the parabola
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Using equations of the tangent and normal expressed by coordinates of the tangency
point and
the figure above;
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a) y-intercept
ct
of the tangent equals half of the ordinates of the tangency point, ct
= y1/2. |
b) the projection
AB
of the segment BP1
of the tangent to the x-axis, i.e., to the axis of the parabola, is
equal to twice the abscissa of the tangency
point, so |
AB
= St
= 2x1 - the line segment
AB
is called the subtangent. |
c) the projection
AC
of the segment CP1
of the normal to the x-axis is equal to the parameter
p, i.e., |
AC
= Sn
= p - the line segment
AC
is called the subnormal. |
As points B
and C
are x-intercepts 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, |
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put
y =
0 into equation of the normal, |
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Thus, the focus
F(p/2,
0) bisects the line segment
BC
whose endpoints are x-intercepts of the tangent and
the normal, as shows the figure above. |
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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. |