# Circle

**Locus of points at a fixed distance,**

*r*, from point

*P*is known as a circle. In this case, distance

**r**is known as radius and point

*P*is centre.

# Equation of a Circle

If the coordinate of the centre is (

Though, equation of circle may not be always given in the above format. Consider the following equation:

It should be written as a summation of (

(

â‡’ (

Hence, this circle has its centre at (1, 2) and its radius = 3 units.

*a*,*b*) and length of radius =*r*, then equation of circle is:(

*x*-*a*)^{2}+ (*y*-*b*)^{2}=*r*^{2}Though, equation of circle may not be always given in the above format. Consider the following equation:

*x*

^{2}+

*y*

^{2}- 2

*x*- 4

*y*- 4 = 0

*x*-*a*)^{2}and (*y*-*b*)^{2}First of all, collect terms of

*x*in one bracket and terms of*y*in other bracket.(

Then start writing the terms in the bracket as squares, keeping the constant terms adjusted.*x*^{2}- 2*x*) + (*y*^{2}- 4*y*) - 4 = 0(

*x*^{2}- 2*x*+ 1) + (*y*^{2}- 4*y*+ 4) - 4 - 5 = 0â‡’ (

*x*- 1)^{2}+ (*y*- 2)^{2}= 3^{2}Hence, this circle has its centre at (1, 2) and its radius = 3 units.

**General Equation of the Circle**

- The general equation of a circle is:
*x*^{2}+*y*^{2}+ 2*gx*+ 2*fy*+*c*= 0*g*, -*f*) and its radius = - If centre of the circle is Origin (0, 0), then equation of circle is:
*x*^{2}+*y*^{2}=*r*^{2}

Example-1

Find the radius and centre of the circle

*x*^{2}+*y*^{2}- 6*x*+ 4*y*- 12 = 0.Solution

Centre of the circle
Comparing this with the equation given in the question:

2
2
Hence, centre = (-
Radius = = = 5 units.

*x*^{2}+*y*^{2}+ 2*gx*+ 2*fy*+*c*= 0 is = (-*g*, -*f*)*g*= -6 â‡’ -*g*= 3*f*= 4 â‡’ -*f*= -2*g*, -*f*) = (3, -2)Example-2

Find the radius and centre of the circle 2

*x*+ 2^{2}*y*- 8^{2}*x*- 7*y*= 0.Solution

We are required to first write it down in the format such that co-efficient of each of
2
In this case,
Hence, centre = (-
Radius = = =

*x*^{2}and*y*^{2}= 1.*x*^{2}+ 2*y*^{2}- 8*x*- 7*y*= 0 can be written as*x*^{2}+*y*^{2}- 4*x*-*y*= 0.*g*= -2 and*f*=*g*, -*f*) =