**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 :__

__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.**

**Equation of a circle under different conditions**

**Equation of a circle under different conditions:**

**(i) Touches both the axes with centre (a, a) and radius a**

**(x−a) ^{2} + (y−a)^{2} = a^{2}**

**(ii) Touches x-axis only with centre (α, a) and radius |a|**

**(x − α) ^{2} + (y−a)^{2} = a^{2}**

**(iii)Touches y–axis only with centre (a, β) and radius |a|**

**(x − a) ^{2} + (y− β)^{2} = a^{2}**

**Example 1. Find the centre and the radius of the circles**

**(i) 3x ^{2} + 3y^{2} − 8x − 10y + 3 = 0.**

**(ii) x ^{2} + y^{2} + 2x sinθ + 2y cosθ − 8 = 0.**

**(iii) 2x ^{2} + λxy + 2y^{2}+ (λ − 4)x + 6y − 5 = 0, for some λ.**

**Solution:(i) We rewrite the given equation as**

**x ^{2} + y^{2} − x − y + 1 = 0 => g = − 4/3, f = − 5/3, c = 1**

**Hence the centre is(4/3 , 5/3) and the radius is**

**√32/9 = 4√2/3 units.**

**(ii) x ^{2} + y^{2} + 2x sin θ + 2y cos θ − 8 = 0.**

**Centre of this circle is (−sin θ, − cos θ)**

**Radius = 3 units.**

**(iii) 2x ^{2} + λxy + 2y^{2} + (λ − 4)x + 6y − 5 = 0**

**rewrite the equation as****Intercepts made by a circle on the axis**

**(i) Length of the intercept made by the circle**

**x ^{2} + y^{2} + 2gx + 2fy + c = 0 on**

**(a) x-axis = AB = 2√(g ^{2} − c)**

**(b) y-axis = CD = 2√(f ^{2} − c)**

**(ii) Intercepts are always positive.**

**(iii) If the circle touches x-axis, then |AB| = 0 ⇒ c = g ^{2}**

**(iv) If the circle touches y-axis, then |CD| = 0 ⇒ c = f ^{2}**

**(v) If the circle touches both the axes, then |CD| = 0 = |AB|**

**⇒ c = g ^{2} = f^{2}.**