What Is the Axis of a Circle and How Is It Related to Diameter and Symmetry
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.
FAQs on Axis of a Circle Explained with Symmetry Concept
1. What is the axis of a circle?
The axis of a circle is any straight line that passes through the centre of the circle and divides it into two equal halves. Every axis of a circle is a line of symmetry.
- A circle has infinitely many axes of symmetry.
- Each axis passes through the centre.
- Each axis divides the circle into two congruent semicircles.
2. How many axes of symmetry does a circle have?
A circle has infinitely many axes of symmetry. Every line passing through the centre of the circle acts as an axis of symmetry.
- If you draw one line through the centre, it divides the circle equally.
- You can rotate that line to any angle through the centre.
- Each such line still divides the circle into two identical halves.
3. Is the diameter the axis of a circle?
Yes, every diameter of a circle is an axis of symmetry. A diameter is a line segment passing through the centre and joining two points on the circle.
- Since it passes through the centre, it divides the circle into two equal semicircles.
- Therefore, each diameter acts as an axis.
- Because there are infinitely many diameters, there are infinitely many axes.
4. What is the difference between the axis and the radius of a circle?
The axis of a circle is a line through the centre dividing the circle into two equal parts, while the radius is a line segment from the centre to the circle’s boundary.
- Axis: Passes completely through the circle.
- Radius: Half of a diameter.
- Radius length = r.
- Diameter length = 2r.
5. What is the equation of the axis of a circle in coordinate geometry?
The equation of an axis of a circle is any straight line passing through the centre (h, k) of the circle. For a circle with equation (x − h)² + (y − k)² = r², any line through (h, k) is an axis.
- Example: If the centre is (2, 3), then
- x = 2 is a vertical axis.
- y = 3 is a horizontal axis.
- Any line like y − 3 = m(x − 2) is also an axis.
6. Why does a circle have infinitely many axes?
A circle has infinitely many axes because it is perfectly symmetrical about every line passing through its centre. The distance from the centre to the boundary is constant in all directions.
- All points on the circle are equidistant from the centre.
- Rotating a line through the centre does not change symmetry.
- Each position of that line forms a new axis.
7. Can you give an example of finding an axis of a circle?
Yes, to find an axis of a circle, first identify its centre and then write any line through it.
- Example: (x − 1)² + (y + 2)² = 16
- Centre = (1, −2)
- One axis: x = 1
- Another axis: y = −2
8. Is the x-axis or y-axis always an axis of a circle?
No, the x-axis or y-axis is an axis of a circle only if the circle’s centre lies on that axis.
- If the centre is (0, 0), both x-axis and y-axis are axes.
- If the centre is (0, 3), only x = 0 (y-axis) is an axis.
- If the centre is (2, 3), neither axis is an axis of symmetry.
9. What is the relationship between the centre and the axis of a circle?
Every axis of a circle must pass through the centre of the circle. The centre is the fixed point that ensures equal division of the circle.
- The centre is equidistant from all points on the circle.
- Any line not passing through the centre cannot divide the circle equally.
- Therefore, the centre lies on every axis of symmetry.
10. What is the difference between the axis of a circle and the axis of a parabola?
The axis of a circle consists of infinitely many lines through its centre, while a parabola has only one axis of symmetry.
- Circle: Infinite axes through the centre.
- Parabola: One axis dividing it into two equal halves.
- Circle equation: (x − h)² + (y − k)² = r².
- Parabola example: y = ax² has axis x = 0.
