The equation of the circle circumscribing the triangle formed by the points (3, 4), (1, 4) and (3, 2) is :

Options

A. `8x^(2)+8y^(2)-16x-13y=0`
B. `x^(2)+y^(2)-4x-8y+19=0`
C. `x^(2)+y^(2)-4x-6y+11=0`
D. `x^(2)+y^(2)-6x-6y+17=0`

Answer

To find the equation of the circle circumscribing the triangle formed by the points (3, 4), (1, 4), and (3, 2), we will follow these steps:

### Step 1: Identify the Points
The points given are:
- A(3, 4)
- B(1, 4)
- C(3, 2)

### Step 2: Write the General Equation of a Circle
The general equation of a circle with center (a, b) and radius r is given by:
\[ (x - a)^2 + (y - b)^2 = r^2 \]

### Step 3: Set Up the Equations
Since the points A, B, and C lie on the circle, we can set up three equations based on the general equation of the circle.

1. For point A(3, 4):
\[ (3 - a)^2 + (4 - b)^2 = r^2 \quad \text{(Equation 1)} \]

2. For point B(1, 4):
\[ (1 - a)^2 + (4 - b)^2 = r^2 \quad \text{(Equation 2)} \]

3. For point C(3, 2):
\[ (3 - a)^2 + (2 - b)^2 = r^2 \quad \text{(Equation 3)} \]

### Step 4: Expand the Equations
Now, we expand each equation:

**Equation 1:**
\[ (3 - a)^2 + (4 - b)^2 = r^2 \\ \Rightarrow (3 - a)^2 + (4 - b)^2 = r^2 \\ \Rightarrow (9 - 6a + a^2) + (16 - 8b + b^2) = r^2 \\ \Rightarrow a^2 + b^2 - 6a - 8b + 25 = r^2 \]

**Equation 2:**
\[ (1 - a)^2 + (4 - b)^2 = r^2 \\ \Rightarrow (1 - a)^2 + (4 - b)^2 = r^2 \\ \Rightarrow (1 - 2a + a^2) + (16 - 8b + b^2) = r^2 \\ \Rightarrow a^2 + b^2 - 2a - 8b + 17 = r^2 \]

**Equation 3:**
\[ (3 - a)^2 + (2 - b)^2 = r^2 \\ \Rightarrow (3 - a)^2 + (2 - b)^2 = r^2 \\ \Rightarrow (9 - 6a + a^2) + (4 - 4b + b^2) = r^2 \\ \Rightarrow a^2 + b^2 - 6a - 4b + 13 = r^2 \]

### Step 5: Equate the Equations
Now, we can equate Equation 1 and Equation 2:
\[ a^2 + b^2 - 6a - 8b + 25 = a^2 + b^2 - 2a - 8b + 17 \]
Cancelling \(a^2\) and \(b^2\) from both sides:
\[ -6a + 25 = -2a + 17 \\ \Rightarrow -4a = -8 \\ \Rightarrow a = 2 \]

Next, equate Equation 1 and Equation 3:
\[ a^2 + b^2 - 6a - 8b + 25 = a^2 + b^2 - 6a - 4b + 13 \]
Cancelling \(a^2\) and \(b^2\) from both sides:
\[ -8b + 25 = -4b + 13 \\ \Rightarrow -4b = -12 \\ \Rightarrow b = 3 \]

### Step 6: Find the Radius
Now we have the center of the circle as (2, 3). To find the radius, substitute \(a\) and \(b\) back into any of the original equations. Let's use Equation 1:
\[ (3 - 2)^2 + (4 - 3)^2 = r^2 \\ \Rightarrow 1 + 1 = r^2 \\ \Rightarrow r^2 = 2 \\ \Rightarrow r = \sqrt{2} \]

### Step 7: Write the Final Equation of the Circle
Now we can write the equation of the circle:
\[ (x - 2)^2 + (y - 3)^2 = 2 \]

### Final Answer
The equation of the circle circumscribing the triangle formed by the points (3, 4), (1, 4), and (3, 2) is:
\[ (x - 2)^2 + (y - 3)^2 = 2 \]