Circle
The circle is a plane figure for which all the points pass through a single plane with equal distance. As it is a plane surface, it's a solid representation of a sphere. The circle has different terminology like radius, diameter, arc, chord circumference, etc. All the terms that arise in the concept of circle, if the circle passes through three points, what is the equation etc. will be studied now.
Circle Passing through 3 Points
If we want to draw a straight line, we need one starting point and one ending point. That means a line needs two points to draw. Similarly, if we are supposed to draw a circle, we need to have some points. But, unlike line segments, we have different ways to draw a circle. Even if a circle can be drawn using a single point because it doesn't have multiple planes, the starting point will also become the ending point.
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Similarly, we can draw a circle using two points. As we had run several circles from a single point, we can also draw several circles from two points. But the current task is to draw a circle passing through three points.
While considering a circle passing through three points, it is important to observe two cases. Because the points may be either collinear or non-collinear, the circle may pass through collinear points or pass through non-colonial points.
The Circle Passes through Collinear Points
Collinear points mean the points which lie on the same line or in the same direction. So by considering these collinear points, if we draw a circle, then the result will be that the circle touches only two points and the third Point May observe either inside of the circle or outside of the circle. In this case, the circle never touches all three points.
The Circle Passes through Three Non-collinear Points.
For this, we need to draw two lines that need to touch the three points. Here it is important to draw two bisectors for two line segments. Now let's take the center and then draw a circle with the center of the circle passing through 3 points. Constituting the equal distance from the center to all sides of the circle, it is known as the radius of a circle passing through 3 points.
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By considering the center and radius of the circle passing through 3 points, it is easy to find the equation of the circle passing through 3 points. Also, the circle passing through collinear points can be stated by proof and theorem.
Find the Equation of the Circle Passing through 3 Points
It is very easy and simple to find the equation of a circle passing through 3 points. For this, we need three non-collier parts with which the circle passes through them.
So let us consider three points initially.
P (\[x_{1}, y_{1}\]), Q (\[x_{2}, y_{2}\]), and R (\[x_{3}, y_{3}\]) are the three points. So, we need to get an equation of a circle passing through these 3 points.
We have a general equation using the two variables.
It is ,\[x^{2} + y^{2} + 2gx + 2fy + c = 0\]
Like the General equation, we need to write equations for three variables with which the circle has to pass through them. Then the obtained equations will be,
\[x{_{1}}^{2} + y{_{1}}^{2} + 2gx_{1} + 2fy_{1} + c = 0\]
\[x{_{2}}^{2} + y{_{2}}^{2} + 2gx_{2} + 2fy_{2} + c = 0\]
and \[x{_{3}}^{2} + y{_{3}}^{2} + 2gx_{3} + 2fy_{3} + c = 0\]
Now, we need to find out the values of g, f, c by solving the three equations of three variables. After getting those values, substitute them in the general equation. The general equation itself is the required equation of the circle passing through three given points.
After getting all the values, we need to draw a circle from that circle. We can easily find the radius of the circle that passes through three points and the center of the circle passes through three points. It is also remembered that the radius should be equal from the center to any point of the circle.
Hence it is the process and explanation of finding the equation of a circle that passes through three points. Those three points should be made collinear because we had already gone through the concept of collinear and non-collinear points.
Creating a 3-Point Circle using a Compass and Straightedge
Join the points to form two lines
Create a perpendicular bisector of one line
Create a perpendicular bisector of another line
Where they cross is the center of the circle
Position the compass in the center, adjust the length to any point, and then draw your circle!
Some Important Key Points from Circles:
Circle: A collection of all points in a plane at a fixed distance from a fixed point. The fixed point is called the center and the fixed distance is called the radius.
Chord: Any part of a line that combines two points in a circle is called a chord. The chord that passes through the center of the circle is called the width.
Secant: A line that crosses a circle by two points is called a secant.
Tangent: A line that crosses a circle only in one place.
Width: The width is twice the radius. It is the longest voice in the circle that passes through the center. All diameters are the same length.
Circular: The circumference of a circle is called the circumference of the circle.
Sector: The area formed by the arc and the two circular radii, by connecting the center to the end of the arc, is called the Field.
Theorem 1: The tangent of any circle point is perpendicular to the radius through the contact area.
Theorem 2: The length of the tents drawn from the outside to the circle is equal.
Some Preparation Tips for Math Students Studying Circles
Statistics is not as challenging as it sounds. For many, it is a matter of dread. However, it is a lesson that will help you even after you leave school! It is also widely used in other subjects such as Physics and Chemistry. The great thing about Math is that, once you get it, it can be your most scoring lesson. Impossible, it is easier than it looks.
Practice as Much as You Can
Statistics are a hands-on topic. You can’t just ‘read’ the chapters, you have to understand the concepts and keep practicing.
Start by Solving Examples
Do not start by solving complex problems. If you have just understood the chapter, solving difficult Math will give you the wrong answer and discourage you. It may make you hate Math even more. Instead, start simple. Solve examples in your textbook.
Clear All Your Doubts
It is easy to cling to doubts in Math. Do not let your doubts form, remove them as soon as possible. The sooner you resolve your doubts, the better off you will be in those articles. Ask your class teacher, friends, or online about the app.
Not All Formulas
When you see something enough, it is registered in your memory, even if it is unconscious. That is why some people choose to attach drawings or formulas to their study table or their room. Make flashcards of all the formulas in your textbook and decorate your room with them, at least until the exam ends!
Understand the Origin
You may think that the release is not so important from a test perspective, but it is important for understanding. You cannot always read the formula, you need to understand the mind behind it.
FAQs on Circle Passing Through 3 Points
1. Find the Equation of the Circle that Passes through Three Points (1, - 6), (2, 1), and (5, 2). Also, Find the Coordinates of its Center and the Length of the Radius.
Let us consider the equation of the required circle be
\[X^{2} + Y^{2} +2gx + 2fy + c = 0\] ……………….(i)
According to the given problem, the above equation passes through the coordinate points (1, - 6), (2, 1), and (5, 2).
Now rewriting the general equation using the given three points, we can get three different equations as follows-
For the point (1, - 6):
1 + 36 + 2g - 12f + c = 0
\[ \Rightarrow 2g - 12f + c = -37.....(ii)\]
For the point (2, 1):
4 + 1 + 4g + 2f + c = 0
\[ \Rightarrow 4g + 2f + c = -5.....(iii)\]
For the point (5, 2):
25 + 4 + 10g + 4f + c = 0
\[ \Rightarrow 10g + 4f + c = -29.....(iv)\]
To get the values of g, f, c we need to solve the obtained three equations.
Subtracting (ii) from (iii) we get,
2g + 14f = 32
\[ \Rightarrow g + 7f = 16.....(v)\]
Again, subtracting (ii) form (IV) we get,
8g + 16f = 8
\[ \Rightarrow g + 2f = 1.....(vi)\]
Now, solving equations (v) and (vi) we get, g = - 5 and f = 3.
Substituting the values of g and f in (iii) we get c = 9.
Therefore, the equation of the required circle which passes through the given tree non-collinear points is
\[x^{2} + y^{2} - 10x + 6y + 9 = 0\]
Then, the coordinates of its center are
(- g, - f) = (5, - 3) and
Radius = (\[\sqrt{g^{2}+f^{2}-c}\])
\[ = \sqrt{25+9-9}\]
\[ = \sqrt{25}\]
= 5 units.
Hence the length of the radius is 5 units.
2. What are the Properties of a Circle?
The properties of your circle are-
The diameter of the circle is twice the radius of the circle. It is also known as the longest chord of the circle.
If the cards and circles are equal, then their circumferences are also equal.
If the radii of two circles are equal, then the circles are said to be congruent to each other.
The distance from the center of the circle to its diameter is always zero.
3. What is the difference between point 2 and point 3 in AutoCAD?
Allows you to create a tangent circle on any of the two objects and define the radius as a third point. The first two points you need to clarify are the bodies or objects in the model space, and the third point can be determined by automatically selecting or adding links. Three collinear points are placed on the same straight line, making it difficult to draw a circle on it. To draw a circle in three points, the points need to be non-collinear and do not lie in the same straight line.
4. What is the importance of a circle in Mathematics?
A circle is defined as a shape that consists of no corners or edges. A circle may be defined as a closed and two-dimensional curved shape in geometry. The point from where all the distances are equal on the circle and the center is called the radius of the circle. Whereas, a semi-circle is half of a complete circle, which is formed by cutting a whole circle along the line segment which passes through the circle center. Therefore, a circle is important for all those reasons which makes a student study the relevant concepts of circles.
