# Standard Equation of a Circle

Expressing a circle in a standard form expression is defined standard equation of a circle.

Imagining a circle in a plane at a particular distance from both axis of the Cartesian coordinate system is the standard form of the circle. Mathematically, a circle can be written in the form of a mathematical expression and it is actually possible by studying the relation of the circle with the Cartesian coordinate system.

## Derivation

Imagine a circle in the Cartesian coordinate system and assume the radius of the circle is

$r$

units. Assume

$P$

is the centre of the circle and it is located at

$a$

units distance in horizontal

$x$

-axis direction and

$b$

units in vertical

$y$

-axis direction from the origin. Therefore, the location of the point

$P$

in the Cartesian coordinate system is

$P\left(a,b\right)$

.

Consider a point on the circle and assume it represents all the points on the circle. It is assumed to call point

$Q$

and the coordinates of the

$Q$

in horizontal and vertical axis direction are

$x$

and

$y$

respectively. Therefore, the coordinates of the point

$Q$

is

$Q\left(x,y\right)$

in the Cartesian coordinate system.

Draw a line from point

$P$

, and it must be parallel to the horizontal axis and draw another line from point

$Q$

, and it should be perpendicular to the same axis and assume they both get intersected each other at a point and it is assumed to call point

$T$

. Thus, a right angled triangle

$\Delta QPT$

is formed inside the circle.

$QT,PT$

and

$PQ$

are opposite side, adjacent side and hypotenuse of the right angled triangle

$\Delta QPT$

.

The length of the opposite side of the right angled triangle

$\Delta QPT$

is

The length of the adjacent side of the right angled triangle

$\Delta QPT$

is

The length of the hypotenuse of the right angled triangle

$\Delta QPT$

is

$PQ=r$

According to Pythagorean theorem, the relation between three sides can be expressed in a mathematical form as given here.

${PQ}^{2}={QT}^{2}+{PT}^{2}$

Substitute lengths of the three sides in this relation to get the equation of a circle in algebraic form expression.

It can be written as follows.

It is an algebraic expression which represents equation of a circle in standard form.

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