# Equation of the Circle when Centre of the Circle coincides with the origin

Expressing a circle in terms of a mathematical equation when the centre of the circle coincides with the origin of the Cartesian coordinate system is defined equation of the circle when the centre of the coincides with the origin.

The centre of the circle may coincide with the origin of the Cartesian coordinate system in a possible case. The circle can be expressed in a mathematical expression form to show how the circle is in Cartesian coordinate system of geometry. It can be developed by understanding the relation of circle with the origin of the coordinate system and it can be expressed in an algebraic form mathematical expression.

## Derivation

Assume, the centre of a circle coincides with the origin of the Cartesian coordinate geometric system. In other words, the origin of the Cartesian coordinate system and centre of the circle are same in this case. Assume, it is denoted by $O$ in the Cartesian coordinate system.

Assume, the radius of the circle is $r$ units.

Consider a point on the circle and assume it represents every point on the circle. Assume, it is located at $x$ units distance in horizontal axis direction and $y$ units distance in vertical axis direction from the origin $O$. So, the location of the point $R$ is written as $R\left(x,y\right)$ in geometric system.

Join point $O$ and point $R$. It is equal to the radius of the circle. Therefore $OR=r$.

Draw a line from point $R$ but the line should be parallel to vertical axis but perpendicular to horizontal axis. Also draw a line from point $O$ on the horizontal axis until the line draw from $R$ intersects it at a point, named point $T$. In this way, a right angled triangle $\Delta ROT$ is constructed within the circle.

According to the right angled triangle $\Delta ROT$

• The line segment $\stackrel{‾}{OR}$ becomes hypotenuse of the right angled triangle $\Delta ROT$ and the length of hypotenuse is $OR=r$
• The line segment $\stackrel{‾}{RT}$ becomes opposite side of the right angled triangle $\Delta ROT$ and the length of the opposite side is $RT=y$
• The line segment $\stackrel{‾}{OT}$ becomes adjacent side of the right angled triangle $\Delta ROT$ and the length of the adjacent side is $OT=x$

The relation between sides of the right angled triangle $\Delta ROT$ can be written in mathematical form by using Pythagorean Theorem.

According to Pythagorean Theorem

${OR}^{2}={OT}^{2}+{RT}^{2}$

Substitute the values of all three sides in this mathematical relation to transform it into equation form.

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

$⇒{x}^{2}+{y}^{2}={r}^{2}$

The algebraic expression is the circle equation when the centre of the circle coincides with the origin of the geometric coordinate system.

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