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.

Save (or) Share
Email subscription
Recommended Math Concepts
How to develop Equation of the Circle when Circle doesn’t passes through the origin and Centre of the Circle lies on y-axis
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Mobile App for Android users
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.