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Circle
and Line
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Mutual position of two circles
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The radical line or the radical
axis
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The pole and the polar
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Angle between two circles
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Mutual position of two circles
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Two circles
k1
and k2
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, |
k1
::
(x
-
p1)2 + (y -
q1)2 -
r12 = 0
and k2
::
(x
-
p2)2 + (y -
q2)2 -
r22 = 0. |
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The radical line or the radical axis |
Subtracting the second equation
from the first gives equation of the line k1-
k2
= 0 through the intersection |
points
A and
B of the circles. |
Coordinates of intersections
A
and B satisfy equations of the circles
k1
and k2 and
the equation of the line k1-
k2
= 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 S1S2, |
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then |
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is
the slope of the radical line. |
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If two circles touch each other outside then, the radical line
is at the same time their common tangent. |
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The pole and the polar
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Given is a circle
k
::
x2
+ y2
= r2 and a point
P(x0,
y0) 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 |
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Therefore, the equation of the circle
k'
is
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or, after squaring and reducing
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k'
:: |
x2
+ y2 - x0
x
- y0
y
= 0. |
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Equation of the line through tangency points, which
is perpendicular to the line OP, is
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p
::
k
- k'
= 0
or
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x0
x
+ y0
y
= r2. |
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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.
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If given is a translated circle
(x
-
p)2 + (y -
q)2 = r2
with the center at the point
S(p,
q), then the equation
of the polar of the point P(x0,
y0) is, |
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(x0 -
p) ·
(x -
p) + (y0 -
q) ·
(y -
q) = r2. |
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Angle between two circles
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Angle between two circles is defined as the angle between the two tangent lines at any of the intersection
points of the circles. |
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Example:
Given are circles
k1
::
x2
+ y2 -
4x -
6y
+ 3 = 0 and
k2
::
x2
+ y2 +
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, |
k1
::
x2
+ y2 -
4x -
6y
+ 3 = 0 (1) |
k2
::
x2
+ y2
+ 6x
+ 4y -
7 = 0 (2) |
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k1-
k2
= 0 =>
-10x
-
10y + 10 = 0
or y
= -x +
1. |
plugging
y
= -x
+ 1 into (1) |
x2
+ (-x +
1)2 -
4x -
6(-x +
1)
+ 3 = 0 |
or 2x2
-
2 = 0,
so x1
=
-1
and
x2 =
1, |
y = -x +
1 =>
y1
=
2
and
y2 =
0. |
thus,
A(-1,
2) and
B(1,
0). |
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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(x0,
y0)
or
A(2,
-2),
S(-3,
1)
and
r2
= 17 |
(x0 -
p) ·
(x -
p) + (y0 -
q) ·
(y -
q) = r2 |
(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 |
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it
follows that, D1(1,
2) and
D2(-2,
-3). |
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