How to Find Equation of a Circle Through Given Three Points - Tutorial

Find Equation of a Circle Through Given Three Points - Definition, Example, Formula

Definition :

A circle is the locus of all points equidistant from a static point, and the equation of a circle is a way to express the definition of a circle on the coordinate plane.

Formula:

r2 = (x - h)2 + (y - k)2 Where, h,k - Center Points of Circle x,y - Circle Coordinates r - Radius
Example :

Find the center point and radius for the equation of a circle passing through (2,2), (2,4) and (5,5) x and y-coordinates.

Given :

A(2,2) B(2,4) C(5,5)

Solution :
Step :1

Substitute the given x and y-coordinates in the circle formula, (2 - h)2 + (2 - k)2 = r2 ------------ (1) (2 - h)2 + (4 - k)2 = r2 ------------ (2) (5 - h)2 + (5 - k)2 = r2 ------------ (3)

Step :2

Let us find the value of k by simplifying the first (1) and second (2) equation, (2 - h)2 + (2 - k)2 = (2 - h)2 + (4 - k)2 4 - 4h + h2+ 4 - 4k + k2 = 4 - 4h + h2+16 - 8k + k2 8 - 4k = 20 - 8k k=3

Step :3

Now, let us find out the value of h by simplifying the second (2) and third (3) equation (2 - h)2 + (2 - k)2 = (5 - h)2 + (5 - k)2 4 - 4h + h2+ 4 - 4k + k2 = 25 - 10h + h2+ 25 - 10k + k2 8 - 4k - 4h = 50 - 10h - 10k 6k + 6h = 42 Substitute k=3 in equation 6h = 24 h=4 Therefore center point is c(h,k) = c(4,3)

Step :4

Substitute h,k values in the given formula r2 = (x - h)2 + (y - k)2 r2 = (2 - 4)2 + (2 - 3)2 r2 = (-2)2 + (-1)2 r2 = 5 r = 2.24

Step :5

Substitute h, k values in the circle formula circle equation = (x - h)2 + (y - k)2 circle equation = (x - 4)2 + (y - 3)2

Result :

Center point is c(h,k) = c(4,3) Radius of a Circle r = 2.24 Circle Equation = (x - 4)2 + (y - 3)2 = (2.24)2

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