Circle


Definition :

        A circle is a set of all points in a plane which are equidistant a fixed point in the plane .

A circle is a closed two-dimensional figure in which the set of all the points in the plane is equidistant from a given point called “centre”. Every line that passes through the circle forms the line of reflection symmetry. Also, it has rotational symmetry around the centre for every angle. The circle formula in the plane is given as:

(x-h)2 + (y-k)2 = r2

where (x,y) are the coordinate points
(h,k) is the coordinate of the centre of a circle
and r is the radius of a circle.

Circle Shaped Objects

There are many objects we have seen in the real world that are circular in shape. Some of the examples are:

  • Ring
  • CD/Disc
  • Bangles
  • Coins
  • Wheels
  • Button
  • Dartboard
  • Hula hoop
Parts of Circle

  • Annulus-The region bounded by two concentric circles. It is basically a ring-shaped object. See the figure below.
  • Arc – It is basically the connected curve of a circle.
  • Sector – A region bounded by two radii and an arc.
  • Segment- A region bounded by a chord and an arc lying between the chord’s endpoints. It is to be noted that segments do not contain the centre.
  • Centre – It is the midpoint of a circle.
  • Chord- A line segment whose endpoints lie on the circle.
  • Diameter- A line segment having both the endpoints on the circle and is the largest chord of the circle.
  • Radius- A line segment connecting the centre of a circle to any point on the circle itself.
  • Secant- A straight line cutting the circle at two points. It is also called an extended chord.
  • Tangent- A coplanar straight line touching the circle at a single point.

Radius of Circle (r)

A line segment connecting the centre of a circle to any point on the circle itself “. The radius of the circle is denoted by “R” or “r”.

Diameter (d) of Circle

A line segment having both the endpoints on the circle. It is twice the length of radius i.e. d = 2r. From the diameter, the radius of the circle formula is obtained as r= d/2.


Circumference (C) of the circle :

The circumference of a circle is defined as the

distance around the circle. The word ‘perimeter’

is also sometimes used, although this usually

refers to the distance around polygons, figures made up of the straight line segment.

A circle circumference formula is given by

  C = πd = 2 π r

Where, π = 3.1415


Area (A) of the circle :

Area of a circle is the amount of space occupied by the circle.

The circle formula to find the area is given by

Area of a circle = πr2


Properties of Circles

  • The outer line of a circle is at equidistant from the centre.
  • The diameter of the circle divides it into two equal parts.
  • Circles which have equal radii are congruent to each other.
  • Circles which are different in size or having different radii are similar.
  • The diameter of the circle is the largest chord and is double the radius.

Difference forms of equation of a circle :

1)  Standard form :
     
The origine , O is the centre of the circle . P(x, y) is any point on the circle . The radius of circle is r 
∴ OP = r
 By section formula 
OP2 = (x -0)2 + (y - 0)2 

∴  we get  r2 = x2 + y2
 ∴    x2 + y2 =   r2
This is the standard equation of circle .

2)  Centre- radius form :
         
Let C(h, k) is the centre and rr is the radius of the circle . P(x, y) is any point on the circle 
CP = r
Also,
∴  CP2 =[ (x - h)2 + (y - k)2]r1/2



r2 = (x - h)2 + (y - k)2
is the centre-radius form of equation of a circle.

3)  Diameter Form :
                  
Let C is the centre A(x1 , y1) , B(x1 , y1) are the end points of a diameter of the circle P(x, y) is any  point on the circle . Angle inscribed in a semi circle is a right angle; hence ∡ APB = 90° , that AP ⊥ BP 
slope of AP = y - y1/x- x1and slope of BP = y -y2/ x- x2. 
 As    AP ⊥ BP , product of their slope is -1
( y - y1)(y -y2) = - (x- x1) (x- x2)
 (x- x1) (x- x2) + ( y - y1)(y -y2) = 0
That is  (x- x1) (x- x2) + ( y - y1)(y -y2) = 0 
This is the diameter form of the equation of circle, where  (x1 , y1) and  (x2 , y2)  are end point of doameter of the circle.

General equation of a circle :
 
    The general equation of a circle is of the form x2 + y2 + 2gx +2 fy + c= 0,   if g2 + f2 - c > 0 
The centre- radius form of equation of a circle is 

r2 = (x - h)2 + (y - k)2
i.e.   x2 - 2hx + h2  + y2 - 2ky + k2  = r2 

i.e. x2 + y2 - 2hx - 2ky + (h2 + k2 - r2) = 0
If this is the same as equation x2 + y2 + 2gx +2 fy + c= 0 , then compairing the coefficients 
2 g = - 2h ,2f = -2k  and c = (h2 + k2 - r2



∴  (h, k) ≡  (-g, -f)  is the centre and 
  r2  = h2 + k2 - c   i.e.  r = ( h2 + k2 - c )1/2  is the radius.
Thus,
 The general equation of a circle x2 + y2 + 2gx +2 fy + c= 0  whose centre is (-g, -f) and radius

 is ( h2 + k2 - c )1/2





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