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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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Parabola and line, examples |
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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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Parabola and line, examples |
Example:
Find the angle between tangents drawn at intersection points of a line and the parabola
y2 = 2px
if the line passes through the focus F(7/4,
0) and its slope
m =
4/3. |
Solution:
As F(p/2,
0)
then, p/2 =
7/4 and the equation
of the parabola y2 =
7x.
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By plugging
m =
4/3 and
F(7/4,
0) into
the equation of the line
y
-
y1 =
m(x
-
x1)
obtained is
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Intersections of the line and the parabola,
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Substituting
coordinates of S1
and S2
in
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we
get the equation of tangents, |
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Slopes of tangents satisfy the perpendicularity
condition, |
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Example:
Find the point on the parabola
y2 = 9x closest to the line
9x + 4y + 24 =
0. |
Solution:
The tangency point of the tangent parallel to
the given line is the closest point.
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9x + 4y +
24 = 0
=> y =
-(9/4)x
-
6, mt =
-
9/4
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The slope of the tangent must satisfy tangency
condition
of the parabola,
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p = 2mc
<= mt =
-
9/4, p
=
9/2
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9/2 = 2 · (-9/4)
· c
=>
c
= -
1
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therefore, the
tangent t
:: y =
-(9/4)x
-
1.
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The solution
to the system of equations of the tangent and the parabola
gives the tangency point, that
is
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College
algebra contents E |
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