CIRCLE

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.

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