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 $P(a,b)$.

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 $Q(x,y)$ 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 $QT=OQ – OT=y – b$

The length of the adjacent side of the right angled triangle $\Delta QPT$ is $QT=OT – OP=x – a$

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.

$r^2={(y – b)}^2+{(x – a)}^2$

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.