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

$ \large \left| \begin{array}{cccc} x^2 + y^2 & x & y & 1 \\ x_1^2 + y_1^2 & x_1 & y_1 & 1 \\ x_2^2 + y_2^2 & x_2 & y_2 & 1 \\ x_3^2 + y_3^2 & x_3 & y_3 & 1 \end{array} \right| = 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}.