

Ellipse
and Line

Angle
between the focal radii at a point of the ellipse

Tangents to an ellipse from a
point outside the ellipse  use of the tangency condition

Construction of tangents from a point outside the ellipse

Ellipse and line examples






Angle between the focal radii at a point of the ellipse

Let
prove that the tangent at a point
P_{1
}of the ellipse
is perpendicular to the bisector of the angle between the focal
radii
r_{1
}and r_{2}.

Coordinates of points,
F_{1}(c,
0),
F_{2}(c,
0) and
P_{1}(x_{1},
y_{1})
plugged into the equation of the line through two given
points determine the lines of the focal radii

r_{1} =
F_{1}P_{1}
and r_{2} =
F_{2}P_{1}, 

and the equation of the tangent at the point
P_{1},





By plugging the slopes of these tree lines into the formula for calculating the angle between lines we find the
exterior angles
j_{1
}and j_{2}
subtended by these lines at P_{1}. 
Thus, using the condition
b^{2}x_{1}^{2}
+ a^{2}y_{1}^{2}
= a^{2}b^{2}, that the point lies on the ellipse, obtained is 

If on the same way we calculate the interior
angle subtended by the focal radii at P_{1},
and which is the supplementary angle of the angle j, 

then compare with the
result which will we obtain by using the doubleangle formula for the angles
j_{1
}and j_{2}, 

To compare obtained results, we multiply both the
numerator and the denominator of the result for
the supplementary angle by
b^{2}, 

what proved the previous statement. 
Therefore, the normal at the point
P_{1}
of the ellipse bisects the interior angle between its focal radii. 

Tangents to an ellipse from a
point outside the ellipse  use of the tangency condition

Coordinates of the point
A(x, y), from which we draw tangents to an ellipse, must satisfy
equations of the tangents, y
= mx + c and their slopes and intercepts,
m and
c, must satisfy the
condition of tangency therefore, using the system of equations, 
(1) y = mx + c
<=
A(x, y) 
(2) a^{2}m^{2} + b^{2}
= c^{2}
determined are equations of the tangents from a point A(x, y)
outside the ellipse. 

Construction of tangents from a point outside the ellipse

With
A
as center, draw an arc through F_{2}, and from
F_{1}
as center, draw an arc of the radius 2a. Tangents are 
then the perpendicular bisectors of the line segments,
F_{2}S_{1} and
F_{2}S_{2}.

We can also draw tangents as lines through A
and the intersection points of the segments F_{1}S_{1} and
F_{1}S_{2}
and the ellipse. 
Thus, these intersections are the tangency points
of the tangents to the ellipse. 
Explanation of the construction lies at the fact that, 
F_{1}S_{1} =
F_{1}D_{1} +
D_{1}S_{1} =
F_{1}D_{1} +
D_{1}F_{2} = 2a

according to the definition of the ellipse, as well as
the point A
is equidistant from points F_{2} and
S_{1},
since the point S_{1}
lies on the arc drawn from A
through F_{2}.





Ellipse and line examples

Example:
At a point A(c,
y
> 0) where c
denotes the focal distance,
on the ellipse
16x^{2}
+ 25y^{2} = 1600
drawn is a tangent to the ellipse, find the area of the triangle that tangent forms by the coordinate axes. 
Solution:
Rewrite the
equation of the ellipse to the standard form 16x^{2}
+ 25y^{2} = 1600  ¸
1600 

We calculate the ordinate of the point
A
by plugging the abscissa into equation of the ellipse

x =
6
=>
16x^{2}
+ 25y^{2} = 1600, 

or, as we know
that the point with the abscise
x = 
c
has the ordinate equal to the value of the
semilatus rectum,






The
area of the triangle formed by the tangent and the coordinate axes, 


Example:
Find a point on the ellipse
x^{2}
+ 5y^{2} = 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.

Rewrite
the equation of the ellipse to determine its axes, 

Tangents and given line have the same slope, so 

Using the tangency condition, determine the intercepts
c, 

therefore, the
equations of tangents,





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 


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 


Example:
At which points
curves, x^{2}
+ y^{2} = 8 and
x^{2}
+ 8y^{2} = 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 

Angle between two curves is the angle between
tangents drawn
to the curves at their point of
intersection.

The tangent to the circle at the intersection
S_{1},

S_{1}(2,
2) =>
x_{1}x
+ y_{1}y =
r^{2},
2x +
2y = 8

or t_{c}_{
}::
y =

x
+ 4
therefore, m_{c}
= 1.

The tangent to the
ellipse at the intersection
S_{1},





The angle between the circle and the ellipse,



Example:
Find the equations of the common tangents of the curves
4x^{2}
+ 9y^{2} = 36 and
x^{2}
+ y^{2} = 5. 
Solution: The common tangents of the ellipse and the circle
must satisfy the tangency conditions of these curves, thus 










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