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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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Example:
Given is the polar
4x + y + 12 =
0 of the parabola
y2 = -4x, find coordinates of the pole and
write equations of the corresponding tangents. |
Solution:
Intersections of the polar and the parabola are the
tangency points of tangents drawn from the pole P.
Thus, by solving the system of equations of the polar and the parabola
we get the tangency points. |
(1) 4x + y +
12 =
0, (1) =>
(2) (-4x
-
12)2 = -4x
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(2) y2 =
-4x
4x2
+ 25x + 36 =
0,
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Equation of the tangent at the point on the parabola,
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The intersection of tangents is the pole P.
Therefore, we solve the system formed by their equations, |
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Pre-calculus contents
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