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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

units. Assume

is the centre of the circle and it is located at

units distance in horizontal

-axis direction and

units in vertical

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

in the Cartesian coordinate system is

.

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

and the coordinates of the

in horizontal and vertical axis direction are

and

respectively. Therefore, the coordinates of the point

is

in the Cartesian coordinate system.

Draw a line from point

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

, 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

. Thus, a right angled triangle

is formed inside the circle.

$QT,PT$

and

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

.

The length of the opposite side of the right angled triangle

is

The length of the adjacent side of the right angled triangle

is

The length of the hypotenuse of the right angled triangle

is

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.

$r^2={(y – b)}^2+{(x – a)}^2$

It can be written as follows.

${(x – a)}^2+{(y – b)}^2=r^2$

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

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