Question

A circle passes through (4, -2) , (-3, 5) , and (6, 2) Using the general form of a circle x 2 y 2 D x E y F = 0 and the points above, fill in missing numbers to create the system of equations. (4, -2) creates D E 1F = (-3, 5) creates D E 1F = (6, 2) creates D E 1F = Solve the system of equations that you created, and write the equation for the circle by filling in the values for D, E, and F. x 2 y 2 x y

Answer 1

To find the equation of the circle that passes through the points (4, -2), (-3, 5), and (6, 2), we can use the general form of a circle equation:

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

Let's create a system of equations using the given points:

(4, -2) creates:
(4)^2 + (-2)^2 + 4D - 2E + F = 0
16 + 4 + 4D - 2E + F = 0
20 + 4D - 2E + F = 0

(-3, 5) creates:
(-3)^2 + (5)^2 - 3D + 5E + F = 0
9 + 25 - 3D + 5E + F = 0
34 - 3D + 5E + F = 0

(6, 2) creates:
(6)^2 + (2)^2 + 6D + 2E + F = 0
36 + 4 + 6D + 2E + F = 0
40 + 6D + 2E + F = 0

Now, we have a system of three equations:

20 + 4D - 2E + F = 0 ...(Equation 1)
34 - 3D + 5E + F = 0 ...(Equation 2)
40 + 6D + 2E + F = 0 ...(Equation 3)

To solve this system of equations, we can use various methods such as substitution, elimination, or matrices. Let's use the elimination method to find the values of D, E, and F.

We'll start by subtracting Equation 1 from Equation 2 and Equation 3:

(Equation 2) - (Equation 1):
(34 - 3D + 5E + F) - (20 + 4D - 2E + F) = 0
34 - 3D + 5E + F - 20 - 4D + 2E - F = 0
-7D + 7E + F = -14 ...(Equation 4)

(Equation 3) - (Equation 1):
(40 + 6D + 2E + F) - (20 + 4D - 2E + F) = 0
40 + 6D + 2E + F - 20 - 4D + 2E - F = 0
2D + 4E = -20 ...(Equation 5)

Now, we have a system of two equations:

-7D + 7E + F = -14 ...(Equation 4)
2D + 4E = -20 ...(Equation 5)

We can solve this system of equations using any desired method, such as substitution or elimination. Let's use substitution to solve for D and E.

From Equation 5, we can express D in terms of E:
2D = -20 - 4E
D = -10 - 2E ...(Equation 6)

Now, substitute Equation 6 into Equation 4:
-7(-10 - 2E) + 7E + F = -14
70 + 14E + 7E + F = -14
21E + F = -84 ...(Equation 7)

Let's solve Equation 7 for F in terms of E:
F = -84 - 21E ...(Equation 8)

Now, we have expressions for D, E, and F in terms of E:
D = -10 - 2E
E = E
F = -84 - 21E

The equation for the circle can be written as:
x^2 + y^2 + Dx + Ey + F = 0
x^2 + y^2 + (-10 - 2E)x + Ey + (-84 - 21E) = 0

Thus, the equation for the circle is:
x^2 + y^2 - 10x - 2Ex + Ey - 84 - 21E = 0.

Please note that the values of D, E, and F depend on the solution to the system of equations.