# Definition

A circle is the locus of a point which moves in a plane such that its distance from a fixed point is always a constant. The fixed point is called the center and the constant distance is called the radius of the circle.

# Circle with center (*h*, *k*) and radius *r*

The equation of circle is (

*x*â€“*h*)^{2}+ (*y*â€“*k*)^{2}=*r*^{2}. In particular, if the center is at the origin, the equation of circle is*x*^{2}+*y*^{2}=*r*^{2}.# General equation of a circle

The general equation of circle is

*x*^{2}+*y*^{2}+ 2*gx*+ 2*fy*+*c*= 0, where*g*,*f*, and*c*are constants. Therefore, coordinates of the center are (â€“*g*, â€“*f*) and radius = .

*Notes:*- A general equation of second degree
*ax*^{2}+ 2*hxy*+*by*^{2}+ 2*gx*+ 2*fy*+*c*= 0 in*x*,*y*represent a circle if- coefficient of
*x*^{2}= coefficient of*y*^{2}, i.e.,*a*=*b* - coefficient of
*xy*is zero, i.e.*h*= 0

- coefficient of
**Concentric circle:**Two circles having the same center*C*(*h*,*k*) but different radii*r*_{1}and*r*_{2}respectively are called concentric circles.**Equation of circle passing through three given points:**The general equation of circle*x*^{2}+*y*^{2}+ 2*gx*+ 2*fy*+*c*= 0 ...(1)*g*,*f*and*c*.*x*_{1},*y*_{1}), (*x*_{2},*y*_{2}), (*x*_{3},*y*_{3}) lie on the circle (1), their coordinates must satisfy its equation. Hence solving equations*g*,*f*, and*c*.**Cyclic quadrilateral:**If all the four vertices of a quadrilateral lie on a circle, then the quadrilateral is called a cyclic quadrilateral. The four vertices are said to be concyclic.- Equation of circle having center (
*Î±*,*Î²*) and touching the*x*-axis is (*x*â€“*Î±*)^{2}+ (*y*â€“*Î²*)^{2}=*Î²*^{2}. - Equation of circle having center (
*Î±*,*Î²*) and touching the*y*-axis is (*x*â€“*Î±*)^{2}+ (*y*â€“*Î²*)^{2}=*Î±*^{2}. - Equation of circle having radius and touching both axes is (
*x*â€“*Î±*)^{2}+ (*y*â€“*Î±*)^{2}=*Î±*^{2}.

# Equation of circle on a given diameter

Let

*A*(*x*_{1},*y*_{1}) and*B*(*x*_{2},*y*_{2}) be the end points of a diameter then (*x*â€“*x*_{1}) (*x*â€“*x*_{2}) + (*y*â€“*y*_{1}) (*y*â€“*y*_{2}) = 0 which is equation for circle in diametric form.Equation of circle passing through two points

*A*and*B*having least radius is the circle having*AB*as diameter.# Parametric form of circle

The parametric coordinates of any point on the circle (

*x*â€“*h*)^{2}+ (*y*â€“*k*)^{2}=*r*^{2}are given by (*h*+*r*cos*Î¸*,*k*+*r*sin*Î¸*), where*Î¸*is parameter (0 â‰¤*Î¸*< 2*Ï€*) then*x*=*h*+*r*cos*Î¸*and*y*=*k*+*r*sin*Î¸*.In particular, coordinates of any point on the circle

*x*^{2}+*y*^{2}=*r*^{2}are (*r*cos*Î¸*,*r*sin*Î¸*) (0 â‰¤*Î¸*< 2*Ï€*).