
Conic
Sections 


Ellipse
and Line

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






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: 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:
The line x
+ 14y

25 = 0 is the polar of the ellipse
x^{2}
+ 4y^{2} = 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, 
(1)
x
+ 14y

25 = 0 =>
x
= 25  14y
=> (2) 
(2) x^{2}
+ 4y^{2} = 25


(25  14y)^{2}
+ 4y^{2} = 25

2y^{2}

7y + 6 = 0,
y_{1} =
3/2 and
y_{2} =
2

y_{1}
and y_{2}
=>
x
= 25  14y,
x_{1} =
4 and
x_{2} =
3.

Thus, the points of
contact D_{1}(4,
3/2) and
D_{2}(3,
2).

The
equations of the tangents at D_{1}
and
D_{2},





The solutions of the system of equations t_{1}
and t_{2
}are the coordinates of the
pole P(1, 7/2). 

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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