Centre of circle passing through A(0,1), B(2,3), C(-2,5) is

Options

A. `(-1,10)`
B. `((-1)/3,10/3)`
C. `(10/3,(-2)/3)`
D. `(1/3,10/3)`

Answer

To find the center of the circle passing through the points A(0,1), B(2,3), and C(-2,5), we can follow these steps:

### Step 1: Write the general equation of a circle
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + C = 0 \]

### Step 2: Substitute point A(0,1) into the equation
Substituting the coordinates of point A into the equation:
\[ 0^2 + 1^2 + 2g(0) + 2f(1) + C = 0 \]
This simplifies to:
\[ 1 + 2f + C = 0 \]
Rearranging gives us:
\[ 2f + C + 1 = 0 \]
Let’s call this Equation (1).

### Step 3: Substitute point B(2,3) into the equation
Substituting the coordinates of point B into the equation:
\[ 2^2 + 3^2 + 2g(2) + 2f(3) + C = 0 \]
This simplifies to:
\[ 4 + 9 + 4g + 6f + C = 0 \]
Rearranging gives us:
\[ 4g + 6f + C + 13 = 0 \]
Let’s call this Equation (2).

### Step 4: Substitute point C(-2,5) into the equation
Substituting the coordinates of point C into the equation:
\[ (-2)^2 + 5^2 + 2g(-2) + 2f(5) + C = 0 \]
This simplifies to:
\[ 4 + 25 - 4g + 10f + C = 0 \]
Rearranging gives us:
\[ -4g + 10f + C + 29 = 0 \]
Let’s call this Equation (3).

### Step 5: Solve for C from Equation (1)
From Equation (1):
\[ C = -2f - 1 \]

### Step 6: Substitute C into Equations (2) and (3)
Substituting \( C = -2f - 1 \) into Equation (2):
\[ 4g + 6f - 2f - 1 + 13 = 0 \]
This simplifies to:
\[ 4g + 4f + 12 = 0 \]
Dividing by 4 gives:
\[ g + f + 3 = 0 \]
Let’s call this Equation (4).

Now substituting \( C = -2f - 1 \) into Equation (3):
\[ -4g + 10f - 2f - 1 + 29 = 0 \]
This simplifies to:
\[ -4g + 8f + 28 = 0 \]
Dividing by 4 gives:
\[ -g + 2f + 7 = 0 \]
Let’s call this Equation (5).

### Step 7: Solve Equations (4) and (5)
Now we have two equations:
1. \( g + f + 3 = 0 \) (Equation 4)
2. \( -g + 2f + 7 = 0 \) (Equation 5)

Adding these two equations:
\[ (g - g) + (f + 2f) + (3 + 7) = 0 \]
This simplifies to:
\[ 3f + 10 = 0 \]
Thus:
\[ f = -\frac{10}{3} \]

### Step 8: Substitute f back to find g
Substituting \( f = -\frac{10}{3} \) into Equation (4):
\[ g - \frac{10}{3} + 3 = 0 \]
This simplifies to:
\[ g = \frac{10}{3} - 3 = \frac{10}{3} - \frac{9}{3} = \frac{1}{3} \]

### Step 9: Find the center of the circle
The center of the circle is given by \( (-g, -f) \):
\[ \text{Center} = \left(-\frac{1}{3}, -\left(-\frac{10}{3}\right)\right) = \left(-\frac{1}{3}, \frac{10}{3}\right) \]

### Final Answer
The center of the circle passing through the points A, B, and C is:
\[ \left(-\frac{1}{3}, \frac{10}{3}\right) \]

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