Circle Passing Through Three points

In this page circle passing through three points we are going to see example problems to find the equation of a circle if three points passing through the circle is given.

Example 1:

Find the equation of a circle passing through the points (0,1) (2,3) and (-2,5).

The general equation of a circle

x² + y² + 2gx + 2fy + c = 0

Now we have to apply these points one by one in the above equation circle passing through three points

equation of a circle which is passing through the point (0,1)

0² + 1² + 2g(0) + 2 f(1) + c = 0

1 + 0 + 2f + c = 0

2f  + c = -1  ----------(1)

equation of a circle which is passing through the point (2,3)

2² + 3² + 2g(2) + 2 f(3) + c = 0

4 + 9 + 4g + 6f + c = 0

4g + 6f  + c = -13 ----------(2)

equation of a circle which is passing through the point (-2,5)

(-2)² + 5² + 2g(-2) + 2 f(5) + c = 0

4 + 25 - 4g + 10f + c = 0

-4g + 10f  + c = -29 ----------(3)

So the three equations are

2f  + c = -1  ----------(1)

4g + 6f  + c = -13 ----------(2)

-4g + 10f  + c = -29 ----------(3)

by solving these three equations we can get the values of f,g and c

4g + 6f  + c = -13 ----------(2)

-4g + 10f  + c = -29 ----------(3)

----------------------

16f + 2c = -42   ---------(5)

Multiplying (1) by 2

4f + 2c = -2

Subtract (5) form this equation

16f + 2c = -42

4f + 2c  = -2

(-)   (-)     (+)

----------------

12f  = -40

f = -40/12

f = -10/3

Substitute f = -10/3 in the first equation

2(-10/3) + c = -1

-20/3 + c = -1

c = -1 + 20/3

c = 17/3

substitute c = 17/3 and f = -10/3 in the second equation

4g + 6(-10/3) + 17/3 = -13

4g - 20 + 17/3 = - 13

4g + (-60+17)/3 = -13

4g - 43/3 = -13

4g = -13 + 43/3

4g = (-39+43)/3

4g = 4/3

g = 4/(4x3)

g = 1/3

Substitute g = 1/3 c = 17/3 ad f = -10/3 in the general equation

x² + y² + 2gx + 2fy + c = 0

x² + y² + 2(1/3)x + 2(-10/3)y + 17/3 = 0

x² + y² + (2/3)x + (-20/3)y + 17/3 = 0

3x² + 3y² + 2x -20y + 17 = 0

So the equation of a circle passing through three points  3x² + 3y² + 2x -20y + 17 = 0

Following are the related topics related to circle passing through three points.

Related Topics

Circle Passing Through Threepoints to Analytical Geometry

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