__Example 5__

Find the equation of the circle such that the three points A(0 , 4), B(3 , 5) and D(7 , 3) are on the circle.
__Solution to Example 5__

The distance from the center C(h , k) of the circle to each of the points A, B and D is constant and equal to the radius r of the circle. Write three equations stating that these distances are equal to the radius r.

d(A,C) = √[(h - 0)^{2} + (k - 4)^{2}] = r

d(B,C) = √[(h - 3)^{2} + (k - 5)^{2}] = r

d(D,C) = √[(h - 7)^{2} + (k - 3)^{2}] = r

Write that d(A,c) = d(B,C) and d(A,C) = d(D,C).

√[(h - 0)^{2} + (k - 4)^{2}] = √[(h - 3)^{2} + (k - 5)^{2}]

√[(h - 0)^{2} + (k - 4)^{2}] = √[(h - 7)^{2} + (k - 3)^{2}]

Square each side of each equation.

(h - 0)^{2} + (k - 4)^{2} = (h - 3)^{2} + (k - 5)^{2}

(h - 0)^{2} + (k - 4)^{2} = (h - 7)^{2} + (k - 3)^{2}

Expand the squares in the above equations and simplify.

-8k + 16 = -6h + 9 -10k + 25

-8k + 16 = -14h + 49 -6k +9

Write the above system of equations in standard form.

2k + 6h = 18

-2k + 14h = 42

Use the method of addition to solve the system.

20h = 60

h = 3

Substitute h by its value 6 in one of the equations to obtain k.

k = 0

We now use one of the distance formula in part a above to find the radius r.

r = √[(3 - 0)^{2} + (0 - 4)^{2}]

= 5

The equation of the circle is given by.

(x - h)^{2} + (y - k)^{2} = r^{2}

(x - 3)^{2} + y^{2} = 25

Shown below is the graph of the circle with the three points.

**Matched Exercise:** Find the equation of the circle such that the three points A(-5 , 0), B(1 , 0) and D(-2 , -3) are on the circle.

Solution

More links and references related to the above topics.

Tutorials on equation of circle.

Tutorials on equation of circle (3).

Match Equations of Circles to Graphs. Excellent interactive activity where equations of circles are matched to graphs.

Interactive tutorial on equation of circle.