
Conic
Sections 


Circle 
General equation of a circle with the center
S(p, q)  translated circle 
Equation of the circle with the
center at the origin O(0, 0)

Circle through three points






Circle 
General equation of a circle with the center
S(p, q)
 translated circle 
A circle with the center
at the point S
(p,
q) and radius r is a set of all points P
(x,
y) of a plane to whom the 
distance from the center,
SP
= r
or 





is
the general equation
of a circle with the
center S(p,
q). 


Equation of the circle with the
center at the origin
O(0,
0),







Example:
A circle passes through points A(2, 4) and
B(2, 6) and its center lies on a line
x + 3y

8 = 0. 
Find equation of the circle. 
Solution:
The intersection of the chord AB
bisector and the given line is the center S
of the circle, since the
bisector is normal through the midpoint M, then 


As the bisector is perpendicular to the line AB 

Equation of the bisector 
M(0,
5) and
m_{n} = 2 =>
y

y_{1} = m( x 
x_{1}), 
gives
y

5 = 2(x 
0)
or 2x
+ y

5 = 0. 






So,
the equation of the circle, (x

p)^{2} + (y 
q)^{2} = r^{2}
=>
(x +
1)^{2} + (y 
3)^{2} = 10. 

Circle through three points

A circle is uniquely determined by three points not lying on the same line. If given are points,
A,
B
and C
then the intersection of any pair of the perpendicular bisectors of the
sides of the triangle ABC is the center of the circle. 
Since all three points lie on the circle, their coordinates must
satisfy equation of the circle 
(x

p)^{2} + (y 
q)^{2} = r^{2}. 
Thus, we obtain the system of three equations in three unknowns
p,
q
and r. 
Subtracting second equation from first and then third from first
we obtain two equations in two unknowns p and
q. 
Solutions of that system plug into any of three equations to
get r. 




Example:
Find equation of a circle passing through three
points, A(2,
6),
B(5, 7)
and C(6, 0). 
Solution:
The coordinates of the points, A,
B
and C
plug into equation (x

p)^{2} + (y 
q)^{2} = r^{2},



Thus,
the equation of the circle through points A,
B
and C,
(x

2)^{2} + (y +
3)^{2} = 25. 









Conic
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