
Conic
Sections 


Ellipse
and Line

Intersection of ellipse and line  tangency condition

Equation of the tangent at a point on the ellipse

Construction of the tangent at a point on 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






Intersection of ellipse and line  tangency condition

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, 
(1) y = mx + c 
(2) b^{2}x^{2} + a^{2}y^{2}
= a^{2}b^{2} 

by plugging (1)
into (2)
=>
b^{2}x^{2}
+ a^{2}(mx + c)^{2} = a^{2}b^{2} 
after rearranging, (a^{2}m^{2}
+ b^{2})·x^{2} + 2a^{2}mc·x
+ a^{2}c^{2} 
a^{2}b^{2} = 0 
obtained
is the quadratic equation in x.
Thus, the coordinate of intersections are, 


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
= a^{2}m^{2}
+ b^{2} 
c^{2}. Thus, if 
D > 0, a line and an ellipse
intersect, 
D = 0,
or 
a^{2}m^{2}
+ b^{2} = c^{2} 
a line is the tangent of the
ellipse and it is tangency
condition. 

The line touches the ellipse at the tangency point whose coordinates are: 

D < 0, a line and an ellipse
do not intersect. 

Equation of the tangent at a point on the ellipse

In the equation
of the line y

y_{1} = m(x

x_{1})
through a given point P_{1}, the slope m
can be determined using known coordinates (x_{1},
y_{1}) of the point of tangency, so 

b^{2}x_{1}x
+ a^{2}y_{1}y
= b^{2}x_{1}^{2}
+ a^{2}y_{1}^{2},
since b^{2}x_{1}^{2}
+ a^{2}y_{1}^{2} =
a^{2}b^{2}
is the condition that P_{1}
lies on the ellipse 

then 
b^{2}x_{1}x
+ a^{2}y_{1}y
= a^{2}b^{2} 
is the
equation of the tangent at the point P_{1}(x_{1},
y_{1})
on the ellipse. 


Construction of the tangent at a point on the ellipse

Draw a circle of a radius
a concentric to the ellipse. Extend the ordinate of the given point to find intersection 
with the circle. The tangent of the circle at
P_{c}
intersects the xaxis at
P_{x}. The tangent to the ellipse at the point
P_{1}on the
ellipse intersects the xaxis at the same point.

To prove this, find the
xintercept of each tangent
analytically. 
Therefore, in both equations of tangents set y = 0 and
solve for x, 

it is the xintercept of the tangent
t_{c}
and the tangent t_{e}.





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 









Intermediate
algebra contents 



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