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 3600 or 2π radians.
Equation of a circle in various forms :
* The simplest equation of the circle is x2 + y2 = r2 whose centre is (0, 0) and radius r.
* The equation (x − a)2 + (y − b)2 = r2 represents a circle with centre (a, b) and radius r.
* The equation x2 + y2 + 2g x + 2f y + c = 0 is the general equation of a circle with centre (−g , −f) and radius √( g2 + f2 − c ) .
Case I: If g2 + f2 − c > 0, then real circle is possible.
Case II: If g2 + f2 − c = 0, then the circle formed is called a point circle.
Case III: If g2 + f2 − c < 0, then no real circle is possible.
* Equation of the circle with points P(x1, y1) and Q(x2, y2) as extremities of a diameter is
(x − x1)(x − x2) + (y − y1)(y − y2) = 0.
* The equation of the circle through three non-collinear points P(x1, y1), Q(x2, y2) and R(x3, y3) is
The general equation of second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a circle, if
* Coefficient of x2 = coefficient of y2 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 = a2
(ii) Touches x-axis only with centre (α, a) and radius |a|
(x − α)2 + (y−a)2 = a2
(iii)Touches y–axis only with centre (a, β) and radius |a|
(x − a)2 + (y− β)2 = a2
Example 1. Find the centre and the radius of the circles
(i) 3x2 + 3y2 − 8x − 10y + 3 = 0.
(ii) x2 + y2 + 2x sinθ + 2y cosθ − 8 = 0.
(iii) 2x2 + λxy + 2y2+ (λ − 4)x + 6y − 5 = 0, for some λ.
Solution:(i) We rewrite the given equation as
x2 + y2 − 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) x2 + y2 + 2x sin θ + 2y cos θ − 8 = 0.
Centre of this circle is (−sin θ, − cos θ)
Radius = 3 units.
(iii) 2x2 + λxy + 2y2 + (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
x2 + y2 + 2gx + 2fy + c = 0 on
(a) x-axis = AB = 2√(g2 − c)
(b) y-axis = CD = 2√(f2 − c)
(ii) Intercepts are always positive.
(iii) If the circle touches x-axis, then |AB| = 0 ⇒ c = g2
(iv) If the circle touches y-axis, then |CD| = 0 ⇒ c = f2
(v) If the circle touches both the axes, then |CD| = 0 = |AB|
⇒ c = g2 = f2.