MTH-5105
Circle
William Low
General Form of the Equation of a Circle
The general form of the equation of a circle with radius r and centred at the origin is:
x2 + y2 = r 2.
The standard form of the equation of a circle with centre (h, k) and radius r is
(x – h)2 + (y – k)2 = r 2.
d=
x2  x1 2  ( y2  y1 ) 2 = x  h2  ( y  k ) 2 = r where (x, y) is any point on the circle.
Note that the parameter h moves the circle in a horizontal direction while the parameter k moves
the circle in a vertical direction. Positive h moves the circle to the right, negative h to the left.
Positive k moves the circle upwards, negative k downwards.
The general form of the equation of a circle is
x2 + y2 + D x + E y + F = 0.
Note: The general form of the equation is written in standard form by completing the square.
The standard equation contains two perfect squares.
Graphing a Relation that Defines a Circle
The relation defined by x2 + y2 + D x + E y + F < 0 is represented by the points situated inside
the circle whose equation is x2 + y2 + D x + E y + F = 0.
The relation defined by x2 + y2 + D x + E y + F > 0 is represented by the points situated outside
the circle whose equation is x2 + y2 + D x + E y + F = 0.
The relation defined by x2 + y2 + D x + E y + F ≤ 0 is represented by the points situated inside
the circle and on the circle whose equation is x2 + y2 + D x + E y + F = 0.
The relation defined by x2 + y2 + D x + E y + F ≥ 0 is represented by the points situated outside
the circle and on the circle whose equation is x2 + y2 + D x + E y + F = 0.
The circle x2 + y2 + D x + E y + F = 0 determines three regions of the plane:
S1 is represented by x2 + y2 + D x + E y + F = 0
S2 is represented by x2 + y2 + D x + E y + F < 0
S3 is represented by x2 + y2 + D x + E y + F > 0
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MTH-5105
Circle
William Low
The domain of a relation is the set of all the first components of the ordered pairs of the relation.
The range of a relation is the set of all the second components of the ordered pairs of the relation.


A subset of real numbers can be expressed in interval form. Thus,  x  R a  x  b  is




represented by ] a , b [ and  x  R a  x  b  corresponds to [ a , b ] .


1.


2.


3.


If the relation R is defined by the equation x2 + y2 + D x + E y + F = 0 or by the
inequality
x2 + y2 + D x + E y + F ≤ 0 , then :


dom R =  x  R h  r  x  h  r   h  r , h  r 




ran R =  x  R k  r  y  k  r   k  r , k  r 


If the relation R is defined by the inequality x2 + y2 + D x + E y + F < 0 , then :


dom R =  x  R h  r  x  h  r   h  r , h  r




ran R =  x  R k  r  y  k  r   k  r , k  r


If the relation R is defined by the inequality x2 + y2 + D x + E y + F > 0 or by the
inequality
x2 + y2 + D x + E y + F ≥ 0 then :
dom R = R
ran R = R
Find the Equation of a Line Tangent to a Circle
A line tangent to a circle touches the circle at the point of tangency. Any tangent to a circle is
perpendicular to the radius which touches the point of tangency.
1
1
and m2 
.
l1 is perpendicular to l2 if and only if m1  m2  1. Hence, m1 
m2
m1
y  y1
Use the formula m  2
to find the slope between two points (x1, y1) and (x2, y2),
x 2  x1
To find the equation of a line given its slope m and the coordinates of one of its points, apply the
y  y1
formula m 
where (x1, y1) are the coordinates of the given point or use the standard
x  x1
equation of a straight line y = mx +b, substituting x1, y1 to find b.
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