SOLUTION: Find the General Equation of the circle 3) The circle passes through (3,0) , (4,2) and (0,1)

Hi, there--

THE PROBLEM:
Find the general equation for the circle that passes through the points (3,0) , (4,2) and (0,1).

A SOLUTION:

The general equation for a circle has the form,

x^2 + y^2 + Dx + Ey + F = 0

We do not know the values of D, E, or F, but we do know three points on the circle. These ordered pairs 
must satisfy our equation.

We will substitute each of the (x,y) pairs into the general equation formula. Then we will have a system of
three equations in three variables (D, E, F). We will solve this system in order to find the equation for 
the circle.

For the ordered pair (3,0): Substitute 3 for x and 0 for y in the general equation
x^2 + y^2 + Dx + Ey + F = 0
(3)^2 + (0)^2 + D(3) + E(0) + F = 0

Simplify.
9 + 3D + F = 0
3D + F = -9

For the ordered pair (4,2):
(4)^2 + (2)^2 + D(4) + E(2) + F = 0

Simplify.
16 + 4 + 4D + 2E + F = 0
4D + 2E + F = -20


For the ordered pair (0,1): 
(0)^2 + (1)^2 + D(0) + E(1) + F = 0

Simplify.
1 + E + F = 0
E + F = -1

Our system of three equations is
3D + F = -9
4D + 2E + F =-20
E + F = -1

Subtract Equation 3 from Equation 1 to eliminate F.
 3D + F = -9
-[E + F = -1]
--------------
3D - E = -8

Solve this equation for E.
E = 3D + 8

Solve the first equation for F.
3D + F = -9
F = -3D - 9

Substitute 3D+8 for E  and -3D-9 for F in the second equation.
4D + 2E + F =-20
4D + 2(3D + 8) + (-3D - 9) = -20

Solve for D.
4D + 6D + 16 - 3D - 9 = -20
7D + 7 = -20
7D = -27
D = -27/7

Substitute -27/7 for D in the first equation.
3D + F = -9
3(-27/7) + F = -9
(-81/7) + F = -9
F = -63/7 + 81/7 
F = 18/7

Substitute 18/7 for F in the third equation
E + F = -1
E + (18/7) = -1
E = -25/7

Substitute -27/7 for D, -25/7 for E, and 18/7 for F in the general equation.
x^2 + y^2 + (-27/7)x + (-25/7)y + (18/7) = 0

The equation in general form is


Check your equation with the three given points. Each pair must make this equation true.
For (3,0):
x^2 + y^2 + (-27/7)x + (-25/7)y + (18/7) = 0
(3)^2 + (0)^2 + (-27/7)(3) + (-25/7)(0) + (18/7) = 0
9 - 81/7 + 18/7 = 0
(63 - 81 +18)/7 = 0 
0/7 = 0 
0 = 0 Check!

For (4,2):
x^2 + y^2 + (-27/7)x + (-25/7)y + (18/7) = 0
(4)^2 + (2)^2 + (-27/7)(4) + (-25/7)(2) + (18/7) = 0
16 + 4 - 108/7 - 50/7 + 18/7 = 0
(140 - 108 - 50 + 18)/7 = 0 
0/7 = 0  
0 = 0 Check!

For (0,1):
x^2 + y^2 + (-27/7)x + (-25/7)y + (18/7) = 0
(0)^2 + (1)^2 + (-27/7)(0) + (-25/7)(1) + (18/7) = 0
1 - 25/7 + 18/7 = 0
(7 - 25 + 18)/7 = 0 
0/7 = 0 
0 = 0 Check!

Hope this helps! Feel free to email if you have any questions about the solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com