To determine the value of \( c \) such that the points \( (2,3) \), \( (0,2) \), \( (4,5) \), and \( (0,c) \) are concyclic, we can follow these steps:
### Step 1: Set up the general equation of a circle
The general equation of a circle can be expressed as:
\[
x^2 + y^2 + 2gx + 2fy + c = 0
\]
where \( g \), \( f \), and \( c \) are constants.
### Step 2: Substitute the first point \( (2,3) \)
Substituting \( (2,3) \) into the circle equation:
\[
(2)^2 + (3)^2 + 2g(2) + 2f(3) + c = 0
\]
This simplifies to:
\[
4 + 9 + 4g + 6f + c = 0
\]
Thus, we have:
\[
4g + 6f + c + 13 = 0 \quad \text{(Equation 1)}
\]
### Step 3: Substitute the second point \( (0,2) \)
Substituting \( (0,2) \) into the circle equation:
\[
(0)^2 + (2)^2 + 2g(0) + 2f(2) + c = 0
\]
This simplifies to:
\[
0 + 4 + 0 + 4f + c = 0
\]
Thus, we have:
\[
4f + c + 4 = 0 \quad \text{(Equation 2)}
\]
### Step 4: Substitute the third point \( (4,5) \)
Substituting \( (4,5) \) into the circle equation:
\[
(4)^2 + (5)^2 + 2g(4) + 2f(5) + c = 0
\]
This simplifies to:
\[
16 + 25 + 8g + 10f + c = 0
\]
Thus, we have:
\[
8g + 10f + c + 41 = 0 \quad \text{(Equation 3)}
\]
### Step 5: Solve the system of equations
We now have three equations:
1. \( 4g + 6f + c + 13 = 0 \)
2. \( 4f + c + 4 = 0 \)
3. \( 8g + 10f + c + 41 = 0 \)
From Equation 2, we can express \( c \) in terms of \( f \):
\[
c = -4f - 4 \quad \text{(Substituting into Equations 1 and 3)}
\]
Substituting \( c \) into Equation 1:
\[
4g + 6f + (-4f - 4) + 13 = 0
\]
This simplifies to:
\[
4g + 2f + 9 = 0 \quad \text{(Equation 4)}
\]
Now substituting \( c \) into Equation 3:
\[
8g + 10f + (-4f - 4) + 41 = 0
\]
This simplifies to:
\[
8g + 6f + 37 = 0 \quad \text{(Equation 5)}
\]
### Step 6: Solve Equations 4 and 5
From Equation 4:
\[
4g + 2f = -9 \quad \Rightarrow \quad 2g + f = -\frac{9}{4} \quad \text{(Equation 6)}
\]
From Equation 5:
\[
8g + 6f = -37 \quad \Rightarrow \quad 4g + 3f = -\frac{37}{2} \quad \text{(Equation 7)}
\]
Now we can solve Equations 6 and 7 simultaneously.
### Step 7: Solve for \( g \) and \( f \)
From Equation 6:
\[
f = -\frac{9}{4} - 2g
\]
Substituting into Equation 7:
\[
4g + 3\left(-\frac{9}{4} - 2g\right) = -\frac{37}{2}
\]
This simplifies to:
\[
4g - \frac{27}{4} - 6g = -\frac{37}{2}
\]
Combining like terms:
\[
-2g - \frac{27}{4} = -\frac{74}{4}
\]
Solving for \( g \):
\[
-2g = -\frac{74}{4} + \frac{27}{4} = -\frac{47}{4}
\]
Thus:
\[
g = \frac{47}{8}
\]
### Step 8: Solve for \( f \) and \( c \)
Substituting \( g \) back into Equation 6:
\[
f = -\frac{9}{4} - 2\left(\frac{47}{8}\right) = -\frac{9}{4} - \frac{47}{4} = -\frac{56}{4} = -14
\]
Now substituting \( f \) back into Equation 2 to find \( c \):
\[
c = -4(-14) - 4 = 56 - 4 = 52
\]
### Step 9: Substitute \( c \) into the equation for the fourth point \( (0,c) \)
Now we substitute \( (0,c) \) into the circle equation:
\[
0 + c^2 + 0 - 19c + 34 = 0
\]
This gives us:
\[
c^2 - 19c + 34 = 0
\]
### Step 10: Solve the quadratic equation
Using the quadratic formula:
\[
c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{19 \pm \sqrt{(-19)^2 - 4 \cdot 1 \cdot 34}}{2 \cdot 1}
\]
Calculating the discriminant:
\[
= \frac{19 \pm \sqrt{361 - 136}}{2} = \frac{19 \pm \sqrt{225}}{2} = \frac{19 \pm 15}{2}
\]
Thus, we have two potential values for \( c \):
\[
c = \frac{34}{2} = 17 \quad \text{or} \quad c = \frac{4}{2} = 2
\]
### Final Answer
Thus, the values of \( c \) for which the points are concyclic are \( c = 2 \) and \( c = 17 \).
