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 .

We have

and

In the right angled triangle

This is the required equation of the circle under given conditions. This form of the circle is known as standard form of the circle.

Conversely, if is any point in the plane satisfying (1), then it is at a distance from . So it is on the circle.

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