To solve the problem, we need to find the equation of the circle that passes through the points A, B, and C, where AB and AC are tangents drawn from point A(0, 1) to the given circle.
### Step 1: Identify the given circle's equation
The equation of the circle is given as:
\[ x^2 + y^2 - 2x + 4y + 1 = 0 \]
### Step 2: Rewrite the circle's equation in standard form
We can rewrite the equation in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \) by completing the square.
1. Rearranging the equation:
\[ x^2 - 2x + y^2 + 4y + 1 = 0 \]
2. Completing the square for \(x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]
3. Completing the square for \(y\):
\[ y^2 + 4y = (y + 2)^2 - 4 \]
4. Substitute back into the equation:
\[ (x - 1)^2 - 1 + (y + 2)^2 - 4 + 1 = 0 \]
\[ (x - 1)^2 + (y + 2)^2 - 4 = 0 \]
\[ (x - 1)^2 + (y + 2)^2 = 4 \]
This shows that the center of the circle is \( (1, -2) \) and the radius is \( 2 \).
### Step 3: Find the equation of the tangents from point A(0, 1)
The formula for the tangents from an external point \( (x_1, y_1) \) to the circle \( (x - h)^2 + (y - k)^2 = r^2 \) is given by:
\[ (x - h)(x_1 - h) + (y - k)(y_1 - k) = r^2 \]
Substituting \( (h, k) = (1, -2) \), \( (x_1, y_1) = (0, 1) \), and \( r = 2 \):
\[ (x - 1)(0 - 1) + (y + 2)(1 + 2) = 4 \]
\[ -(x - 1) + 3(y + 2) = 4 \]
\[ -x + 1 + 3y + 6 = 4 \]
\[ -x + 3y + 7 = 4 \]
\[ -x + 3y = -3 \]
\[ x - 3y = 3 \] (This is one tangent)
### Step 4: Find the second tangent
Using the same method, we can find the second tangent:
1. The equation will be similar, but we will have to consider the other tangent direction.
2. The second tangent can be derived similarly, resulting in another linear equation.
### Step 5: Find the equation of the circle passing through A, B, and C
The circle passing through points A, B, and C will have its center at the midpoint of the line segment joining the center of the original circle and point A. The diameter will be along the line connecting A and the center of the original circle.
Using the midpoint formula:
\[ \text{Midpoint} = \left( \frac{0 + 1}{2}, \frac{1 - 2}{2} \right) = \left( \frac{1}{2}, -\frac{1}{2} \right) \]
The radius will be the distance from this midpoint to point A.
### Step 6: Write the equation of the new circle
Using the center and radius, we can write the equation of the new circle:
\[ \left( x - \frac{1}{2} \right)^2 + \left( y + \frac{1}{2} \right)^2 = r^2 \]
### Step 7: Compare with given options
After simplifying the equation, we can compare it with the provided options to find the correct one.
### Final Answer
The equation of the circle passing through points A, B, and C is:
\[ x^2 + y^2 - x + y - 2 = 0 \]
