Equation of a Circle when centre of the Circle lies on x-axis

Expressing a circle in the form of a mathematical expression when the centre of the circle lies on the horizontal axis and the circle does not pass through the origin of the Cartesian coordinate system is defined as equation of a circle when the centre of the circle lies on $x$-axis.

The centre of the circle may lie on the horizontal $x$-axis of the Cartesian coordinate system without passing through the origin. The circle can be written in an equation form to reveal the relationship of the circle with Cartesian coordinate system and it also reveals how the circle exactly is in the coordinate geometric system. The relation of the circle with coordinate system is useful to develop an equation of the circle when the centre of the circle lies on the $x$-axis.

Derivation

Consider a circle which has $r$ units’ radius in the Cartesian coordinate system and assume, the centre of a circle is lied on the horizontal $x$-axis. Assume, it is called as point $P$ and also assume it is located at a distance of $a$ units from the origin in horizontal $x$-axis. The point $P$ is actually lied on the $x$-axis. So, the vertical distance of the point $P$ from origin is zero. Therefore, the coordinates of the point $P$ is $P\left(a,0\right)$.

Consider one point on the circle and it is assumed to call point $R$. Assume, the point $R$ is located at $x$ units distance from origin in horizontal axis direction and y units distance from origin in vertical axis direction. Therefore, the location of the point $R$ in the Cartesian coordinate system is $R\left(x,y\right)$.

Draw a line from point $R$ towards horizontal axis perpendicularly and simultaneously draw a line from point $P$ on the horizontal axis. Finally, the two lines perpendicularly get intersected at a point, called $Q$. Thus, a right angled triangle $\Delta RPQ$ is constructed inside the circle.

According to right angled triangle $\Delta RPQ$, the relation between three sides of the triangle can be written in mathematical form as per the Pythagorean Theorem.

${PR}^{2}={PQ}^{2}+{QR}^{2}$

The length of the each side of the right angled triangle can be determined geometrically.

• Length of the opposite side from the right angled triangle is $RQ=y$.
• Length of the adjacent side from the right angled triangle is
• Length of the hypotenuse from the right angled triangle is $PR=r$

The mathematical relation, expressed based on the Pythagorean Theorem can be transformed as an algebraic expression by substituting these three values of the sides of the right angled triangle.

This algebraic equation represents a circle, when circle’s centre lies on the horizontal axis. It can be written as follows.

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