
Conic
Sections 


Ellipse
and Line

Polar and pole of the ellipse

Equation of the polar of the given
point

Ellipse and line examples






Polar and pole of the ellipse

If from a point A(x_{0}, y_{0}), exterior to the ellipse, drawn are tangents, then the secant line passing through the
contact points, D_{1}(x_{1}, y_{1}) and
D_{2}(x_{2}, y_{2}) is the polar of the point
A. The point
A is called the
pole of the 
polar,
as shows the right figure. 
Coordinates of the point
A(x_{0}, y_{0}) must satisfy the
equations of tangents, thus 
t_{1} ::
b^{2}x_{0}x_{1}
+ a^{2}y_{0}y_{1}
= a^{2}b^{2} 
t_{2} ::
b^{2}x_{0}x_{2}
+ a^{2}y_{0}y_{2}
= a^{2}b^{2} 
and after
subtracting
t_{1
}
t_{2}

b^{2}x_{0}(x_{2}

x_{1})
+ a^{2}y_{0}(y_{2}

y_{1})
= 0

obtained
is 






the slope of the secant line through points of contact
D_{1}and
D_{2}.

Thus, the equation of the secant line


and after rearranging 

b^{2}x_{0}x
+ a^{2}y_{0}y
= b^{2}x_{0}x_{1}
+ a^{2}y_{0}y_{1},
since b^{2}x_{0}x_{1}
+ a^{2}y_{0}y_{1}
= a^{2}b^{2} 
follows 
b^{2}x_{0}x
+ a^{2}y_{0}y
= a^{2}b^{2} 
the
equation of
the polar p
of the
point A(x_{0}, y_{0}). 


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:
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 










Intermediate
algebra contents 



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