Axis of a Circle

A circle is an area where a moving point in a plane is located, with its distance from a fixed point in the plane being constant at all times. The circle's radius and fixed point are referred to as its centre and radius, respectively. One of the two axes that splits the coordinate system into two equally sized halves is the x-axis. Here we will learn about the axis of a circle and how to find axis of a circle.

Equation of a Circle:

A circle represents a collection of points whose distance from a particular point is constant. This fixed point is called the circle's centre. It is described by a constant known as the circle's radius, abbreviated r.

A circle represents a collection of points whose distance from a particular point is constant. This fixed point is called the circle's centre. It is described by a constant known as the circle's radius, abbreviated r. So kids, the general equation for a circle is:

${(x - {x_1})^2} + {(y - {y_1})^2} = {r^2}$

Equation of Normal to the Circle:

The given normal runs between the circle's centre and the point (x1, y1), or (0, 0). Now, all we have to do to get the normal equation is apply the two-point version of the equation of a straight line.

The equation for normal to the circle will be:

$\frac{y}{{{y_1}}} = \frac{x}{{{x_1}}}$

Standard Equation of Circle:

The second-degree equation with the variables x and y is used to express the equation of a unit circle. Circle's standard equation is represented as:

${(x - a)^2} + {(y - b)^2} = {r^2}$

Where the point of the circle (a, b) is the centre of the circle and r is the circle's radius.

Equation of a Circle Whose Centre Lies on X-Axis:

If the centre of a circle lies on the x-axis, say at a point (a, 0), then the equation of the circle will be:

${x^2} + {y^2} - 2ax + {a^2} - {r^2} = 0$

The centre of the circle is represented by the point (a, b), while the equation represents the radius of the circle ${(x - a)^2} + {(y - b)^2} = {r^2}$. Now, because the centre is on the x-axis, the values of b must be zero. As a result, the centre of such a circle will be shown as:

${(x - a)^2} + {(y - b)^2} = {r^2}$

Substituting b = 0 in the above equation,

${(x - a)^2} + {(y)^2} = {r^2}$

Expanding the equation further, we get

${x^2} + {y^2} - 2ax + {a^2} - {r^2} = 0$

Thus, the equation of a circle whose centre lies on x axis will be:

${x^2} + {y^2} - 2ax + {a^2} - {r^2} = 0$

Examples on Axis of a Circle:

Example 1:

Find the equation of a circle whose centre is on the x-axis at -3 and whose radius is 6 units.

Solution:

Radius of circle = 6 units

The y coordinate of the centre will be zero because the circle's centre lies on the x-axis.

The required equation of the circle will be:

$\begin{array}{l}{(x + 3)^2} + {y^2} = {6^2}\\{x^2} + 6x + 9 + {y^2} = 36\\{x^2} + {y^2} + 6x - 27 = 0\end{array}$

Example 2:

Find the equation of a circle whose centre is on the x-axis at 5 and whose radius is 4 units.

Solution:

Radius of circle = 4 units

The y coordinate of the centre will be zero because the circle's centre lies on the x-axis.

The required equation of the circle will be:

$\begin{array}{l}{(x - 5)^2} + {y^2} = {4^2}\\{x^2} - 10x + 25 + {y^2} = 16\\{x^2} + {y^2} - 10x + 25 - 16 = 0\\{x^2} + {y^2} - 10x + 9 = 0\end{array}$

Example 3:

Find the equation of a circle of radius 5 whose centre lies on x – the axis and passes through the point (2, 3).

Solution:

We are aware that a circle's standard equation is provided by ${(x - a)^2} + {(y - b)^2} = {r^2}$ with the circle's radius being r and its centre being the point (a, b). The centre is located on the x-axis, passes through the coordinates (2, 3), and has a radius of 5.

Hence,

Radius=5

$\begin{array}{l}\sqrt {{{(a - 2)}^2} + {{(0 - 3)}^2}} = 5\\{(a - 2)^2} + 9 = 25\\a - 2 = \pm 4\\a = 6\\a = - 2\end{array}$

Conclusion:

In this article, we learnt about the definition of a circle, the standard equation of the circle and the equation of the circle whose centre lies on the x-axis. Some examples are also seen in the article to clear the concept thoroughly. We have a basic knowledge of circles and topics related to them, and we can deal with the problems related to circles and solve them easily in day-to-day life.