How to Find the Equation of a Circle Passing Through Three Points
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: Complete Guide
1. What is the fundamental condition for a unique circle to pass through three points?
For a unique circle to pass through three points, the points must be non-collinear. This means they cannot all lie on the same straight line. If the points are non-collinear, they form a triangle, and a unique circle, known as the circumcircle, can be drawn passing through all three vertices.
2. Why is it impossible to draw a circle that passes through three collinear points?
It is impossible because the centre of any circle passing through two points must lie on the perpendicular bisector of the line segment connecting them. If you have three collinear points (A, B, and C), the perpendicular bisector of AB will be parallel to the perpendicular bisector of BC. Since parallel lines never intersect, there is no single point that can serve as the centre for a circle passing through all three points simultaneously.
3. How do you find the equation of a circle passing through three non-collinear points using algebra?
To find the equation of a circle passing through three points, you can use the general form of a circle's equation: x² + y² + 2gx + 2fy + c = 0. The process involves these steps:
Substitute the coordinates (x, y) of each of the three given points into the general equation.
This creates a system of three linear equations with three variables: g, f, and c.
Solve this system of equations simultaneously to find the unique values for g, f, and c.
Substitute these values back into the general equation to get the final equation of the circle.
4. What is the step-by-step geometric construction for drawing a circle through three points?
The geometric method to draw a circle through three non-collinear points A, B, and C involves finding its centre. Here are the steps:
Join the points to form two line segments, for example, AB and BC.
Construct the perpendicular bisector for the line segment AB.
Construct the perpendicular bisector for the line segment BC.
The point where these two perpendicular bisectors intersect is the centre (O) of the required circle.
Set the compass width to the distance from the centre O to any of the three points (e.g., OA). This distance is the radius (r). Draw the circle.
5. How is the circumcircle of a triangle related to a circle passing through three points?
They are essentially the same concept. When three points are non-collinear, they can be considered the vertices of a triangle. The circumcircle of a triangle is defined as the unique circle that passes through all three of its vertices. Therefore, finding the circle that passes through three given non-collinear points is equivalent to finding the circumcircle of the triangle formed by those points.
6. Once you have the equation x² + y² + 2gx + 2fy + c = 0, how do you find the circle's centre and radius?
After finding the values of g, f, and c, you can directly determine the circle's properties from the general equation. The coordinates of the centre of the circle are given by (-g, -f). The radius (r) of the circle is calculated using the formula: r = √(g² + f² - c). This provides a direct algebraic method to find the geometric properties of the circle.
7. Can two different circles pass through the same three non-collinear points? Explain why.
No, only one unique circle can pass through the same three non-collinear points. The reason lies in the uniqueness of the circle's centre and radius. As established by the geometric construction, the centre is the single intersection point of the perpendicular bisectors of the chords connecting the points. Since two distinct lines can intersect at only one point, there is only one possible centre. Consequently, the radius (the distance from this centre to any of the points) is also fixed, defining a single, unique circle.
