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.
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 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, so the circle may pass through collinear points or pass through non-colonial 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.
For this, we need to draw two lines which need to touch the three points. Here it is important to draw two bisectors for two line segments. Now let's take a centre and then draw a circle with the centre of a circle passing through 3 points. Constituting the equal distance from the centre 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 centre 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 passes through collinear points can be stated by proof and theorem.
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 (x1, y1), Q (x2, y2), and R (x3, y3) 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 , x2 + y2 + 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,
x12 + y12 + 2gx1 + 2fy1 + c = 0
x22 + y22 + 2gx2 + 2fy2 + c = 0
and x32 + y32 + 2gx3 + 2fy3 + 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 centre of the circle passes through three points. It is also remembered that the radius should be equal from the centre 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.
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 Centre and the Length of the Radius.
Let us consider the equation of the required circle be
x² + y² + 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
⇒ 2g - 12f + c = -37 ……………….(ii)
For the point (2, 1):
4 + 1 + 4g + 2f + c = 0
⇒ 4g + 2f + c =- 5 ……………….(iii)
For the point (5, 2):
25 + 4 + 10g + 4f + c = 0
⇒ 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
⇒ g + 7f = 16 ……………….(v)
Again, Subtracting (ii) form (iv) we get,
8g + 16f = 8
⇒ 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
x² + y² - 10x + 6y + 9 = 0
Then, the coordinates of its centre are
(- g, - f) = (5, - 3) and
radius = g²+f²−c−−−−−−−−−√
= 5 units.
Hence the length of the radius is 5 units.
2. What are the Properties of a Circle?
Ans. 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 centre of the circle to its diameter is always zero.