
Conic
Sections 


Circle
and Line

Mutual position of two circles

The radical line or the radical
axis

The pole and the polar

Angle between two circles






Mutual position of two circles

Two circles
k_{1}
and k_{2}
intersect if the distance between their centers is less than the sum, but greater than
difference, of their radii. 
The coordinates of the intersection points of two circles
we calculate by solving their equations as system of two quadratic
equations, 
k_{1}
::
(x

p_{1})^{2} + (y 
q_{1})^{2} 
r_{1}^{2 } = 0
and k_{2}
::
(x

p_{2})^{2} + (y 
q_{2})^{2} 
r_{2}^{2 }= 0. 

The radical line or the radical axis 
Subtracting the second equation
from the first gives equation of the line k_{1}
k_{2}
= 0 through the intersection 
points
A and
B of the circles. 
Coordinates of intersections
A
and B satisfy equations of the circles
k_{1}
and k_{2} and
the equation of the line k_{1}
k_{2}
= 0. 
This line is called the
radical line and represents the
locus or set of all points in the plane of equal power
with respect to two nonconcentric circles. 
The radical line
is a line perpendicular to the line connecting the centers of
the two circles. 
Since
the slope of the line S_{1}S_{2}, 


then 

is
the slope of the radical line. 




If two circles touch each other outside then, the radical line
is at the same time their common tangent. 

The pole and the polar

Given is a circle
k
::
x^{2}
+ y^{2}
= r^{2 }and a point
P(x_{0},
y_{0}) outside the circle. The contact
points of tangents from P
to the circle, are at the same time the intersections of
the given circle k and the circle
k' whose center is the midpoint of the line
segment OP,
that is 

Therefore, the equation of the circle
k'
is


or, after squaring and reducing


k'
:: 
x^{2}
+ y^{2}  x_{0}
x
 y_{0}
y
= 0. 


Equation of the line through tangency points, which
is perpendicular to the line OP, is

p
::
k
 k'
= 0
or

x_{0}
x
+ y_{0}
y
= r^{2}. 


This line is called the
polar
of the
point P
with respect to the circle, and point P
is called the
pole of the polar.





If given is a translated circle
(x

p)^{2} + (y 
q)^{2} = r^{2}
with the center at the point
S(p,
q), then the equation
of the polar of the point P(x_{0},
y_{0}) is, 

(x_{0} 
p) ·
(x 
p) + (y_{0} 
q) ·
(y 
q) = r^{2}. 



Angle between two circles

Angle between two circles is defined as the angle between the two tangent lines at any of the intersection
points of the circles. 

Example:
Given are circles
k_{1}
::
x^{2}
+ y^{2} 
4x 
6y
+ 3 = 0 and
k_{2}
::
x^{2}
+ y^{2} +
6x
+ 4y 
7 = 0,
find their intersections. 
Solution:
Subtracting given equations of circles gives equation of the line through their intersection points called the radical line or radical axis, 
k_{1}
::
x^{2}
+ y^{2} 
4x 
6y
+ 3 = 0 (1) 
k_{2}
::
x^{2}
+ y^{2}
+ 6x
+ 4y 
7 = 0 (2) 

k_{1}
k_{2}
= 0 =>
10x

10y + 10 = 0
or y
= x +
1. 
plugging
y
= x
+ 1 into (1) 
x^{2}
+ (x +
1)^{2} 
4x 
6(x +
1)
+ 3 = 0 
or 2x^{2}

2 = 0,
so x_{1}
=
1
and
x_{2} =
1, 
y = x +
1 =>
y_{1}
=
2
and
y_{2} =
0. 
thus,
A(1,
2) and
B(1,
0). 




Example:
From the point
A(2,
2)
drawn are tangents to the circle
(x +
3)^{2} + (y 1)^{2} =
17, find equations of
tangents using the polar and taking the point A
as the pole. 
Solution:
Coordinates of the point A
plug into equation of the polar, 
so,
P(x_{0},
y_{0})
or
A(2,
2),
S(3,
1)
and
r^{2}
= 17 
(x_{0} 
p) ·
(x 
p) + (y_{0} 
q) ·
(y 
q) = r^{2} 
(2 +
3)
·
(x +
3) + (2 
1) ·
(y 
1) =
17, 
which
gives, 
p
::
5x

3y
+ 1
= 0 
the equation of the polar. 
By solving system of equations of the polar and the circle
we calculate coordinates of points of contact, 
(1) 5x

3y
+ 1
= 0 
(2) (x +
3)^{2} + (y 1)^{2} =
17 

it
follows that, D_{1}(1,
2) and
D_{2}(2,
3). 












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