
Conic
Sections 


Circle
and Line

Tangents to a circle from a point outside the circle  use of the
tangency condition

Angle between a line and a circle

Mutual position of two circles






Tangents to a circle from a point outside the circle  use of the
tangency condition

Example:
Find the angle between tangents drawn from the point
A(1,
7) to the circle
x^{2}
+ y^{2} = 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) r^{2}·(m^{2}
+ 1)
= c^{2} 

25(m^{2}
+ 1)
= (m + 7)^{2} 
12m^{2
}
7m

12 = 0 => m_{1}
= 
3/4
and
m_{2}
= 4/3 
c_{1}
= 
3/4 + 7 = 25/4 and
c_{2}
= 4/3 + 7 = 25/3 
therefore,
the equations of tangents, 
t_{1}
::
y =

(3/4)x + 25/4 and
t_{2}
::
y =
(4/3)x + 25/3. 
Slopes of tangents satisfy perpendicularity condition, that is

m_{1}
= 
1/m_{2}
=>
j =
90°. 




Example:
Given is a line
3x +
y + 1 = 0 and a circle
x^{2}
+ y^{2} 
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
m_{t} =
1/3 
x^{2}
+ y^{2} 6x
4 y
+ 3 = 0
=>
(x

3)^{2} + (y 
2)^{2} = 10 
thus,
S(3,
2)
and
r^{2}
= 10. 
To
find intersections c
we use the tangency condition, 
r^{2}·(m^{2}
+ 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 c_{1}
= 1/3
and c_{2}
= 19/3. 
The equations of
tangents are, 
t_{1}
::
y =
(1/3)x
1/3 and t_{2}
::
y =
(1/3)x
+19/3. 




Example: Find equations of the common tangents to
circles x^{2}
+ y^{2} =
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, 

The
equation 10
+ 2m 
c = +3c
does not satisfy given conditions. 
The
equation 10
+ 2m 
c = 3c
or c = 5 
m plugged into (1) 

Therefore,
the equations of the common tangents are, 


Angle between a line and a circle

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. 

Example:
Find the angle between a line
2x +
3y 
1 = 0 and a circle
x^{2}
+ y^{2} +
4x
+ 2y  15
= 0. 
Solution:
Coordinates of
intersections of the line and the circle calculate by solving the system, 

Rewrite
the equation of the circle to standard form, 
x^{2}
+ y^{2} +
4x
+ 2y  15
= 0
=>
(x

p)^{2} + (y 
q)^{2} = r^{2} 
thus,
(x +
2)^{2} + (y + 1)^{2} = 20,
S(2,
1) and
r^{2}
= 20. 



Equation of the tangent at the intersection
S_{1}, 
S_{1}(4,
3)
=>
(x_{1} 
p) ·
(x 
p) + (y_{1} 
q) ·
(y 
q) = r^{2}

(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



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. 









College
algebra contents E 



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