Equations of Circle in various forms

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 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

= 0.

Notes:

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.

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