Answers to Exercise in Tutorial on Equation of Circle(2)

A detailed solution to the matched exercise in Tutorial on Equation of Circle(2) is presented.


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


Let (h , k) be the center and r be the radius of the circle and write the general equation of the circle as follows
(x - h)2 + (y - k)2 = r2
The three points are on the circle and therefore satisfy the above equation of the circle. Hence the three equations obtained by substituting the values of the coordinates x and y of the points into the equation
(-5 - h)2 + (0 - k)2 = r2
(1 - h)2 + (0 - k)2 = r2
(- 2 - h)2 + (- 3 - k)2 = r2
We now need to solve the above non linear system of equations in three variable. Expand the squares in the above equations and rewrite them as follows
25 + 10 h + h2 + k2 = r2     (1)
1 - 2 h + h2 + k2 = r2     (2)
4 + 4h + h2 + 9 + 6 k + k2 = r2     (3)
Subtract the left and right hand sides of equations (1) and (2) to obtain another equivalent equation
(25 + 10 h + h2 + k2) - (1 - 2 h + h2 + k2) = r2 - r2
Simplify to obtain a simpler equation
25 + 10 h - 1 + 2h = 0
Solve the above linear equation to obtain
h = - 2
Subtract the left and right hand sides of equations (2) and (3) to obtain another equivalent equation
(1 - 2 h + h2 + k2) - (4 + 4h + h2 + 9 + 6 k + k2) = r2 - r2
Simplify to obtain the linear equation
- 6 h - 6 k - 12 = 0
Substitute h by -2 in the above equation and solve for k
- 6 (-2) - 6 k - 12 = 0
k = 0
Substitute h by - 2 and k by 4 in any of the three equations (1), (2) or (3) and solve for r. Let us use equation (1)
25 + 10 (-2) + (-2)2 + (0)2 = r2
r2 = 9
r = 3
Substitute h, k and r by their values found above into the general equation to obtain the equation of the circle through the three given points.
(x + 2)2 + y2 = 32

In the figure below are shown the given circle and the given three points and we can see that the points are on the circle.
Circle through three given points.

Figure 1. Graph of the given circle through the given through points.

More References and links

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.