# Equation of a Circle

## Equation of a Circle Introduction

The circle is a two-dimensional plane shape in which all the points are placed at a fixed distance from a fixed point called the centre. Since all the points are placed at a fixed distance, a circle can be mathematically represented by using equations. Therefore, the equation that describes a circle is called the equation of the circle.

## Equation of a Circle in General Form or Cartesian Form

### Equation of a Circle when the Centre of the circle is at the origin

To understand the concept of the equation of a circle, let us take an example of a circle as shown in the figure. In the figure, the centre of the circle is placed at the origin $O$ of the $x-y$ cartesian coordinate system and the radius of the circle is $10$ units.

Now, let us take a point $P$ which is on the circle as shown in the figure. The coordinate of the point $P$ is $(x, y)$.

If a line is drawn from the point $P$ to $Q$ so that $PQ \bot OQ$, then it forms a right-angled triangle $\vartriangle OPQ$.

From Pythagoras theorem,

${\left( {OP} \right)^2} = {\left( {OQ} \right)^2} + {\left( {PQ} \right)^2}$

From figure, $OP=10$, $OQ=x$ and $PQ=y$

${10^2} = {x^2} + {y^2}$

Since $10$ is the radius in the above equation, therefore, the standard from can be written as,

${x^2} + {y^2} = {r^2}$

Where, $x$ and $y$ are the coordinate of the point $P$ and $r$ is the radius of the circle. There is an infinite number of points in a circle. For all the points, the equation of the circle remains the same.

### Equation of a Circle when the Centre of the circle is not at the origin

Now, the question is, what if the centre of the circle is not at the origin? How to write the equation of a circle when the centre of the circle is a point say $C$?

To understand the concept, let us consider a circle as shown in the figure. The centre of the circle is at point $C$. The coordinate of the point $C$ is $(a, b)$. Let the point $A$ is on the circle and coordinate of $A$ is $(x, y)$. The radius of the circle is $10$ units.

Now, if a line $AB$ is drawn as shown in the figure such that $AB \bot CD$, then $\vartriangle ABC$ is a right-angled triangle.

From figure,

Hypotenuse of the $\vartriangle ABC$ = Radius of the Circle = $10$ units

Base of the $\vartriangle ABC$ = $x-a$

Perpendicular of the $\vartriangle ABC$ = $y-b$

As per Pythagoras Theorem,

${\left( {AC} \right)^2} = {\left( {AB} \right)^2} + {\left( {BC} \right)^2}$

${10^2} = {\left( {y - b} \right)^2} + {\left( {x - a} \right)^2}$

Rearranging,

${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {10^2}$

Where, $10$ is the radius of the circle. Therefore, if the radius of the circle is denoted by $r$, then the standard form can be written as,

${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {r^2}$

Which is the equation of the circle when the circle is placed at any point other than origin.

## Parametric Form

The equation of a circle can also be represented in parametric form using trigonometric functions sine and cosine as given below:

### When the Centre of the circle is placed at the Origin

$x = r\cos t$

$y = r\sin t$

Where, $r$ is the radius, $t$ is the angle between the points $(a, b)$ and $(x, y)$ as shown in the figure.

### When the Centre of the circle is not at the origin

$x = a + r\cos t$

$y = b + r\sin t$

## Equation of a Circle in different forms

The equation of a circle in different forms are tabulated below:

The equation of a circle when the circle is placed at the origin of the x-y plane is ${x^2} + {y^2} = {r^2}$
When the centre of a circle is at the origin of x-y plane then the equation of the circle is ${x^2} + {y^2} = {r^2}$