Expressing a circle in terms of a mathematical form expression when the centre of the circle lies on the y-axis is defined as equation of a circle when centre of the circle lies on $y$-axis.

Geometrically, centre of a circle can be lied on the vertical axis without passing through the origin of the Cartesian coordinate system. The circle can be written in a mathematical expression to understand this case in mathematics. It can be developed as per the relation of the circle with the Cartesian coordinate system.

Imagine a circle which have radius of $r$ units and assume the centre of the circle lies on the vertical axis and also assume it does not pass through the origin. Assume, the centre of the circle is called as point $P$ and it is located at a distance of $b$ units from the origin of the Cartesian coordinate system. The centre of the circle is lied on the y-axis. So, the horizontal distance from centre of the circle to origin is zero. Therefore, the centre of the circle is located at $P(0,b)$.

Consider a point on the circle and assume it is called point $R$. Actually, it is a point which represents every one of the points on the circle. Assume it is located at $x$ units and $y$ units distance in horizontal and vertical axis from the origin of the Cartesian coordinate system. Therefore, the point $R$ is located at $R(x,y)$ in the Cartesian coordinate system.

Now, draw a line from point $P$ and it should be parallel to the horizontal axis, and draw another line from point $R$ and it should be parallel to the vertical axis and they both perpendicularly intersect at a point which is assumed to call point $Q$. Thus, a right angled triangle $\Delta RPQ$ is formed inside the circle.

In this right angled triangle, $RQ,PQ$ and $PR$ are opposite side, adjacent side and hypotenuse respectively. According to the Pythagorean Theorem, the sides of the right angled triangle can be expressed in a mathematical form as follows.

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

The lengths of the right angled triangle can be evaluated from the right angled triangle.

Length of the opposite side is $RQ=OR\u2013OQ=y\u2013b$

Length of the adjacent side is $PQ=x$

Length of the hypotenuse is $PR=r$

Substitute these values in the mathematical relation of the three sides.

${r}^{2}={x}^{2}+{(y\u2013b)}^{2}$

$\Rightarrow {x}^{2}+{(y\u2013b)}^{2}={r}^{2}$

This algebraic equation is the required circle equation when the circle’s centre lies on the vertical axis. It can expand further as given below.

$\Rightarrow {x}^{2}+{y}^{2}\u20132yb+{b}^{2}\u2013{r}^{2}=0$

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