**DEFINITION **

**A circle is the locus of a point which moves in such a way that its distance from a fixed point, called the centre, is always a constant **. The distance r from the centre is called the radius of the circle.

Twice the radius is known as the diameter d = 2r

The perimeter C of a circle is called the circumference, and is given by

**C = πd = 2πr.**

The angle a circle subtends from its centre is a full angle equal to 360^{0} or 2π radians.

__Equation of a circle in various forms :__

* The simplest equation of the circle is x^{2} + y^{2} = r^{2} whose centre is (0, 0) and radius r.

* The equation (x − a)^{2} + (y − b)^{2} = r^{2} represents a circle with centre (a, b) and radius r.

* The equation x^{2} + y^{2} + 2g x + 2f y + c = 0 is the general equation of a circle with centre (−g , −f) and radius √( g^{2} + f^{2} − c ) .

**Case I:** If g^{2} + f^{2} − c > 0, then **real circle** is possible.

**Case II:** If g^{2} + f^{2} − c = 0, then the circle formed is called a **point circle.**

**Case III:** If g^{2} + f^{2} − c < 0, then **no real circle** is possible.

* Equation of the circle with points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) as extremities of a diameter is

**(x − x _{1})(x − x_{2}) + (y − y_{1})(y − y_{2}) = 0.**

* The equation of the circle through three non-collinear points P(x_{1}, y_{1}), Q(x_{2}, y_{2}) and R(x_{3}, y_{3}) is

= 0.

**Notes:**

The general equation of second degree ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 represents a circle, if

* Coefficient of x^{2} = coefficient of y^{2} i.e. a = b

* Coefficient of xy = zero i.e. h = 0.