To find the equation of the circle passing through the points (12, 43), (18, 39), and (42, 3), we can start with the general equation of a circle:
\[
x^2 + y^2 + 2gx + 2fy + c = 0
\]
Here, \(g\), \(f\), and \(c\) are constants that we need to determine. Since we have three points, we can substitute each point into the equation to form a system of equations.
### Step 1: Substitute the first point (12, 43)
Substituting \(x = 12\) and \(y = 43\):
\[
12^2 + 43^2 + 2g(12) + 2f(43) + c = 0
\]
Calculating \(12^2\) and \(43^2\):
\[
144 + 1849 + 24g + 86f + c = 0
\]
This simplifies to:
\[
24g + 86f + c + 1993 = 0 \quad \text{(Equation 1)}
\]
### Step 2: Substitute the second point (18, 39)
Substituting \(x = 18\) and \(y = 39\):
\[
18^2 + 39^2 + 2g(18) + 2f(39) + c = 0
\]
Calculating \(18^2\) and \(39^2\):
\[
324 + 1521 + 36g + 78f + c = 0
\]
This simplifies to:
\[
36g + 78f + c + 1845 = 0 \quad \text{(Equation 2)}
\]
### Step 3: Substitute the third point (42, 3)
Substituting \(x = 42\) and \(y = 3\):
\[
42^2 + 3^2 + 2g(42) + 2f(3) + c = 0
\]
Calculating \(42^2\) and \(3^2\):
\[
1764 + 9 + 84g + 6f + c = 0
\]
This simplifies to:
\[
84g + 6f + c + 1773 = 0 \quad \text{(Equation 3)}
\]
### Step 4: Solve the system of equations
Now we have three equations:
1. \(24g + 86f + c + 1993 = 0\)
2. \(36g + 78f + c + 1845 = 0\)
3. \(84g + 6f + c + 1773 = 0\)
We can eliminate \(c\) by subtracting Equation 1 from Equation 2:
\[
(36g + 78f + c + 1845) - (24g + 86f + c + 1993) = 0
\]
This gives:
\[
12g - 8f - 148 = 0 \quad \Rightarrow \quad 3g - 2f = 37 \quad \text{(Equation 4)}
\]
Next, subtract Equation 2 from Equation 3:
\[
(84g + 6f + c + 1773) - (36g + 78f + c + 1845) = 0
\]
This gives:
\[
48g - 72f - 72 = 0 \quad \Rightarrow \quad 2g - 3f = 3 \quad \text{(Equation 5)}
\]
### Step 5: Solve Equations 4 and 5
From Equation 4:
\[
3g = 2f + 37 \quad \Rightarrow \quad g = \frac{2f + 37}{3}
\]
Substituting \(g\) into Equation 5:
\[
2\left(\frac{2f + 37}{3}\right) - 3f = 3
\]
Multiplying through by 3 to eliminate the fraction:
\[
2(2f + 37) - 9f = 9
\]
Expanding and simplifying:
\[
4f + 74 - 9f = 9 \quad \Rightarrow \quad -5f + 74 = 9 \quad \Rightarrow \quad -5f = -65 \quad \Rightarrow \quad f = 13
\]
Now substituting \(f = 13\) back into Equation 4:
\[
3g - 2(13) = 37 \quad \Rightarrow \quad 3g - 26 = 37 \quad \Rightarrow \quad 3g = 63 \quad \Rightarrow \quad g = 21
\]
### Step 6: Find \(c\)
Now substituting \(g\) and \(f\) back into Equation 1 to find \(c\):
\[
24(21) + 86(13) + c + 1993 = 0
\]
Calculating:
\[
504 + 1118 + c + 1993 = 0 \quad \Rightarrow \quad c + 3615 = 0 \quad \Rightarrow \quad c = -3615
\]
### Final Equation of the Circle
Now we have \(g = 21\), \(f = 13\), and \(c = -3615\). The equation of the circle is:
\[
x^2 + y^2 + 42x + 26y - 3615 = 0
\]
### Step 7: Verify the additional points
Now we need to check if the points \((-54, -69)\) and \((-81, -38)\) satisfy the equation of the circle.
#### For point (-54, -69):
Substituting \(x = -54\) and \(y = -69\):
\[
(-54)^2 + (-69)^2 + 42(-54) + 26(-69) - 3615 = 0
\]
Calculating:
\[
2916 + 4761 - 2268 - 1794 - 3615 = 0
\]
This simplifies to:
\[
0 = 0 \quad \text{(True)}
\]
#### For point (-81, -38):
Substituting \(x = -81\) and \(y = -38\):
\[
(-81)^2 + (-38)^2 + 42(-81) + 26(-38) - 3615 = 0
\]
Calculating:
\[
6561 + 1444 - 3402 - 988 - 3615 = 0
\]
This simplifies to:
\[
0 = 0 \quad \text{(True)}
\]
### Conclusion
The equation of the circle is:
\[
x^2 + y^2 + 42x + 26y - 3615 = 0
\]
And it passes through the points \((-54, -69)\) and \((-81, -38)\).
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