The equation of radical axis of two circles is `x + y = 1`. One of the circles has the ends ofa diameter at the points `(1, -3) and (4, 1)` and the other passes through the point (1, 2).Find the equations of these circles.

Explanation

To find the equations of the two circles based on the given information, we can follow these steps:

### Step 1: Determine the center and radius of the first circle
The first circle has its diameter endpoints at the points (1, -3) and (4, 1).

1. **Find the center**: The center of the circle is the midpoint of the diameter.
\[ \text{Center} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{1 + 4}{2}, \frac{-3 + 1}{2} \right) = \left( \frac{5}{2}, -1 \right) \]

2. **Find the radius**: The radius is half the distance between the endpoints of the diameter.
\[ \text{Distance} = \sqrt{(4 - 1)^2 + (1 - (-3))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, the radius \( r = \frac{5}{2} \).

3. **Equation of the first circle**: The standard form of the circle's equation is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting the center and radius:
\[ \left(x - \frac{5}{2}\right)^2 + (y + 1)^2 = \left(\frac{5}{2}\right)^2 \]
Expanding this gives:
\[ \left(x - \frac{5}{2}\right)^2 + (y + 1)^2 = \frac{25}{4} \]

### Step 2: Expand the equation of the first circle
Expanding the equation:
\[ \left(x^2 - 5x + \frac{25}{4}\right) + (y^2 + 2y + 1) = \frac{25}{4} \]
Combining terms:
\[ x^2 + y^2 - 5x + 2y + 1 = 0 \]
This is the equation of the first circle.

### Step 3: Set up the equation for the second circle
The second circle passes through the point (1, 2). We will use the general form of the circle's equation:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
Substituting the point (1, 2):
\[ 1 + 4 + 2g + 4f + c = 0 \implies 2g + 4f + c = -5 \quad \text{(Equation 1)} \]

### Step 4: Use the radical axis equation
The radical axis equation is given as \( x + y - 1 = 0 \). This can be rewritten as:
\[ x + y = 1 \]
This implies:
\[ 2g + 5 = 2f - 2 = c - 1 \]
From this, we can derive:
1. \( 2g + 5 = 2f - 2 \)
2. \( 2g + 5 = c - 1 \)

### Step 5: Solve for \( g, f, c \)
From the first equation:
\[ 2f = 2g + 7 \implies f = g + \frac{7}{2} \]
From the second equation:
\[ c = 2g + 6 \]

### Step 6: Substitute back into Equation 1
Substituting \( f \) and \( c \) into Equation 1:
\[ 2g + 4\left(g + \frac{7}{2}\right) + (2g + 6) = -5 \]
This simplifies to:
\[ 2g + 4g + 14 + 2g + 6 = -5 \implies 8g + 20 = -5 \implies 8g = -25 \implies g = -\frac{25}{8} \]
Now substituting \( g \) back to find \( f \) and \( c \):
\[ f = -\frac{25}{8} + \frac{7}{2} = -\frac{25}{8} + \frac{28}{8} = \frac{3}{8} \]
\[ c = 2\left(-\frac{25}{8}\right) + 6 = -\frac{50}{8} + \frac{48}{8} = -\frac{2}{8} = -\frac{1}{4} \]

### Step 7: Write the equation of the second circle
Now substituting \( g, f, c \) into the general circle equation:
\[ x^2 + y^2 - \frac{25}{4}x + \frac{3}{4}y - \frac{1}{4} = 0 \]
Multiplying through by 4 to eliminate fractions:
\[ 4x^2 + 4y^2 - 25x + 3y - 1 = 0 \]

### Final Equations
1. First Circle: \( x^2 + y^2 - 5x + 2y + 1 = 0 \)
2. Second Circle: \( 4x^2 + 4y^2 - 25x + 3y - 1 = 0 \)