# Circles, Objectives, Definition of the Circle, Equation of a Circle, when Coordinates of the Centre and Radius Are Given

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Notice the path in which the tip of the hand of a watch moves. (See Fig.)

Again, notice the curve traced out when a nail is fixed at a point and a thread of certain length is tied to it in such a way that it can rotate about it, and on the other end of the thread a pencil is tied. Then move the pencil around the fixed nail keeping the thread in a stretched position (See Fig)

Certainly, the curves traced out in the above examples are of the same shape and this type of curve is known as a circle.

The distance between the tip of the pencil and the point, where the nail is fixed is known as the radius of the circle.

## Objectives

After studying this lesson, you will be able to:

Derive and find the equation of a circle with a given centre and radius;

state the conditions under which the general equation of second degree in two variables represents a circle;

Derive and find the centre and radius of a circle whose equation is given in general form;

Find the equation of a circle passing through:

Three non-collinear points

Two given points and touching any of the axes;

## Definition of the Circle

A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point in the same plane remains constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.

## Equation of a Circle

Can we find a mathematical expression for a given circle?

Let us try to find the equation of a circle under various given conditions.

## When Coordinates of the Centre and Radius Are Given

Let be the centre and be they radius of the circle. Coordinates of the centre are given to be , say.

Take any point on the circle and draw perpendiculars and on . Again, draw perpendicular to .