This is a continuation of tutorial on equations of circles.
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) = sqrt[(h  0)^{2} + (k  4)^{2}] = r
d(B,C) = sqrt[(h  3)^{2} + (k  5)^{2}] = r
d(D,C) = sqrt[(h  7)^{2} + (k  3)^{2}] = r

Write that d(A,c) = d(B,C) and d(A,C) = d(D,C).
sqrt[(h  0)^{2} + (k  4)^{2}] = sqrt[(h  3)^{2} + (k  5)^{2}]
sqrt[(h  0)^{2} + (k  4)^{2}] = sqrt[(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 = sqrt[(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.
Answer.
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