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Conic
Sections |
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Circle
and Line
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Tangents to a circle from a point outside the circle - use of the
tangency condition
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Angle between a line and a circle
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Mutual position of two circles
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Tangents to a circle from a point outside the circle - use of the
tangency condition
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Example:
Find the angle between tangents drawn from the point
A(-1,
7) to the circle
x2
+ y2 = 25 . |
Solution:
Equations of tangents we
find from the system formed by equation of the line and the
tangency condition, |
A(-1,
7) => (1)
y =
mx + c => c =
m + 7
=> (2) |
(2) r2·(m2
+ 1)
= c2 |
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25(m2
+ 1)
= (m + 7)2 |
12m2
-
7m
-
12 = 0 => m1
= -
3/4
and
m2
= 4/3 |
c1
= -
3/4 + 7 = 25/4 and
c2
= 4/3 + 7 = 25/3 |
therefore,
the equations of tangents, |
t1
::
y =
-
(3/4)x + 25/4 and
t2
::
y =
(4/3)x + 25/3. |
Slopes of tangents satisfy perpendicularity condition, that is
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m1
= -
1/m2
=>
j =
90°. |
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Example:
Given is a line
-3x +
y + 1 = 0 and a circle
x2
+ y2 -
6x
- 4y
+ 3 = 0,
find equations of tangents to the circle which are perpendicular to the line. |
Solution:
Slopes of tangents are determined by condition of
perpendicularity, therefore |
y =
3x -
1,
m =
3
so that
mt =
-1/3 |
x2
+ y2 -6x
-4 y
+ 3 = 0
=>
(x
-
3)2 + (y -
2)2 = 10 |
thus,
S(3,
2)
and
r2
= 10. |
To
find intersections c
we use the tangency condition, |
r2·(m2
+ 1)
= (q -m
p -
c)2 |
10[(-1/3)2
+ 1]
= [2 -(-1/3)·3
-
c]2 or
(3
-
c)2
= 100/9 |
(3
-
c)
= ± 10/3
so c1
= -1/3
and c2
= 19/3. |
The equations of
tangents are, |
t1
::
y =
-(1/3)x
-1/3 and t2
::
y =
-(1/3)x
+19/3. |
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Example: Find equations of the common tangents to
circles x2
+ y2 =
13 and (x
+ 2)2 + (y +
10)2 =
117. |
Solution:
Slopes and intersections of common tangents to the circles must satisfy tangency condition of both
circles. Therefore, values for slopes m
and intersections c
we calculate from the system of equations, |
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The
equation -10
+ 2m -
c = +3c
does not satisfy given conditions. |
The
equation -10
+ 2m -
c = -3c
or c = 5 -
m plugged into (1) |
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Therefore,
the equations of the common tangents are, |
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Angle between a line and a circle
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The angle between a line and a circle is the angle formed by the line and the tangent to the circle at the
intersection point of the circle and the given line. |
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Example:
Find the angle between a line
2x +
3y -
1 = 0 and a circle
x2
+ y2 +
4x
+ 2y - 15
= 0. |
Solution:
Coordinates of
intersections of the line and the circle calculate by solving the system, |
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Rewrite
the equation of the circle to standard form, |
x2
+ y2 +
4x
+ 2y - 15
= 0
=>
(x
-
p)2 + (y -
q)2 = r2 |
thus,
(x +
2)2 + (y + 1)2 = 20,
S(-2,
-1) and
r2
= 20. |
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Equation of the tangent at the intersection
S1, |
S1(-4,
3)
=>
(x1 -
p) ·
(x -
p) + (y1 -
q) ·
(y -
q) = r2
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(-4
+
2)
·
(x +
2)
+ (3 + 1)
·
(y
+
1)
= 20
=>
t
::
-
x
+ 2y
-
10
= 0
or y =
1/2x + 5 |
The angle between the line and the circle is the angle formed by the line and the tangent to the circle at the
intersection point, therefore
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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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