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Ellipse
and Line
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Intersection of ellipse and line - tangency condition
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Equation of the tangent at a point on the ellipse
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Construction of the tangent at a point on the ellipse
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Angle between the focal radii at a point of the ellipse
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Ellipse and line examples
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| Intersection of ellipse and line - tangency condition
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| Common points of a line and an ellipse
we find by solving their equations as a system of two equations in two
unknowns, x
and y, |
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(1) y = mx + c |
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(2) b2x2 + a2y2
= a2b2 |
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by plugging (1)
into (2)
=>
b2x2
+ a2(mx + c)2 = a2b2 |
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after rearranging, (a2m2
+ b2)·x2 + 2a2mc·x
+ a2c2 -
a2b2 = 0 |
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is the quadratic equation in x.
Thus, the coordinate of intersections are, |
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Using the above solutions follows that a line and an ellipse can have one of three possible mutual positions
depending of the value of the discriminant D
= a2m2
+ b2 -
c2. Thus, if |
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D > 0, a line and an ellipse
intersect, |
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D = 0,
or |
a2m2
+ b2 = c2 |
a line is the tangent of the
ellipse and it is tangency
condition. |
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The line touches the ellipse at the tangency point whose coordinates are: |
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D < 0, a line and an ellipse
do not intersect. |
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| Equation of the tangent at a point on the ellipse
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| In the equation
of the line y
-
y1 = m(x
-
x1)
through a given point P1, the slope m
can be determined using known coordinates (x1,
y1) of the point of tangency, so |
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| b2x1x
+ a2y1y
= b2x12
+ a2y12,
since b2x12
+ a2y12 =
a2b2
is the condition that P1
lies on the ellipse |
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then |
b2x1x
+ a2y1y
= a2b2 |
is the
equation of the tangent at the point P1(x1,
y1)
on the ellipse. |
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| Construction of the tangent at a point on the ellipse
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| Draw a circle of a radius
a concentric to the ellipse. Extend the ordinate of the given point to find intersection |
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with the circle. The tangent of the circle at
Pc
intersects the x-axis at
Px. The tangent to the ellipse at the point
P1on the
ellipse intersects the x-axis at the same point.
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To prove this, find the
x-intercept of each tangent
analytically. |
Therefore, in both equations of tangents set y = 0 and
solve for x, |
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it is the x-intercept of the tangent
tc
and the tangent te.
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| Angle between the focal radii at a point of the ellipse
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prove that the tangent at a point
P1
of the ellipse is perpendicular to the bisector of the angle between the focal radii
r1
and r2.
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Coordinates of points,
F1(-c,
0),
F2(c,
0) and
P1(x1,
y1) plugged
into the equation of the line through two given points determine the lines of the focal radii
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| r1 =
F1P1
and r2 =
F2P1, |
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and the equation of the tangent at the point
P1,
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By plugging the slopes of these tree lines into the formula for calculating the angle between lines we find the
exterior angles
j1
and j2
subtended by these lines at P1. |
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b2x12
+ a2y12
= a2b2, that the point lies on the ellipse, obtained is |
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| If on the same way we calculate the interior
angle subtended by the focal radii at P1,
and which is the supplementary angle of the angle j, |
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| then compare with the
result which will we obtain by using the double-angle formula for the angles
j1
and j2, |
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| To compare obtained results, we multiply both the
numerator and the denominator of the result for
the supplementary angle by
b2, |
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| what proved the previous statement. |
| Therefore, the normal at the point
P1
of the ellipse bisects the interior angle between its focal radii. |
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| Ellipse and line examples
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| Example:
Find a point on the ellipse
x2
+ 5y2 = 36 which is the closest, and which is the farthest from the
line 6x + 5y - 25 =
0. |
Solution:
The tangency
points of tangents to the ellipse which are parallel with the given
line are, the
closest and the farthest points from the line.
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| Rewrite
the equation of the ellipse to determine its axes, |
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| Tangents and given line have the same slope, so |
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| Using the tangency condition, determine the intercepts
c, |
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therefore, the
equations of tangents,
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Solutions of the system of equations of tangents to the ellipse determine the points of contact, i.e., the
closest and the farthest point of the ellipse from the given line, thus |
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| Example: Determine equation of the ellipse which the line
-3x +
10y = 25 touches at the point
P(-3,
8/5). |
| Solution:
As the given line is the tangent to the ellipse,
parameters, m
and c
of the line must satisfy the tangency condition, and the point P
must satisfy the equations of the line and the ellipse, thus |
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| Example:
At which points
curves, x2
+ y2 = 8 and
x2
+ 8y2 = 36, intersect? Find the angle between two
curves. |
| Solution:
Given curves are the circle and the
ellipse. The solutions
of the system of their equations determine the intersection points, so |
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Angle between two curves is the angle between
tangents drawn
to the curves at their point of
intersection.
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The tangent to the circle at the intersection
S1,
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S1(2,
2) =>
x1x
+ y1y =
r2,
2x +
2y = 8
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or tc
::
y =
-
x
+ 4
therefore, mc
= -1.
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The tangent to the
ellipse at the intersection
S1,
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| The angle between the circle and the ellipse,
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| Example:
The line x
+ 14y
-
25 = 0 is the polar of the ellipse
x2
+ 4y2 = 25. Find coordinates of the pole. |
| Solution:
Intersections of the polar and the ellipse are points
of contact of tangents drawn from the pole P
to ellipse, thus solutions of the system of equations, |
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x
+ 14y
-
25 = 0 =>
x
= 25 - 14y
=> (2) |
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(2) x2
+ 4y2 = 25
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(25 - 14y)2
+ 4y2 = 25
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2y2
-
7y + 6 = 0,
y1 =
3/2 and
y2 =
2
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y1
and y2
=>
x
= 25 - 14y,
x1 =
4 and
x2 =
-3.
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Thus, the points of
contact D1(4,
3/2) and
D2(-3,
2).
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The
equations of the tangents at D1
and
D2,
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The solutions of the system of equations t1
and t2
are the coordinates of the
pole P(1, 7/2). |
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| Example:
Find the equations of the common tangents of the curves
4x2
+ 9y2 = 36 and
x2
+ y2 = 5. |
| Solution: The common tangents of the ellipse and the circle
must satisfy the tangency conditions of these curves, thus |
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