Find the equation of the circle passing throught (1,1) and the points of intersection of the circles `x^(2)+y^(2)+13x-3y=0` and `2x^(2)+2y^(2)+4x-7y-25=0`

Answer

To find the equation of the circle passing through the point (1, 1) and the points of intersection of the circles given by the equations \(x^2 + y^2 + 13x - 3y = 0\) and \(2x^2 + 2y^2 + 4x - 7y - 25 = 0\), we can follow these steps:

### Step 1: Rewrite the equations of the circles
The first circle's equation is:
\[ x^2 + y^2 + 13x - 3y = 0 \]
The second circle's equation can be simplified by dividing everything by 2:
\[ x^2 + y^2 + 2x - \frac{7}{2}y - \frac{25}{2} = 0 \]

### Step 2: Find the points of intersection of the two circles
To find the points of intersection, we can eliminate one variable by subtracting the first equation from the second. Let's rearrange both equations:
1. \(x^2 + y^2 + 13x - 3y = 0\) (Equation 1)
2. \(x^2 + y^2 + 2x - \frac{7}{2}y - \frac{25}{2} = 0\) (Equation 2)

Subtract Equation 1 from Equation 2:
\[ (2x - 13x) + (-\frac{7}{2}y + 3y) - \frac{25}{2} = 0 \]
This simplifies to:
\[ -11x + \left(3 - \frac{7}{2}\right)y - \frac{25}{2} = 0 \]
\[ -11x - \frac{1}{2}y - \frac{25}{2} = 0 \]
Multiplying through by -2 to eliminate the fraction:
\[ 22x + y + 25 = 0 \quad \text{(Equation 3)} \]

### Step 3: Substitute Equation 3 into one of the original circle equations
We can substitute \(y = -22x - 25\) from Equation 3 into Equation 1:
\[ x^2 + (-22x - 25)^2 + 13x - 3(-22x - 25) = 0 \]
Expanding this:
\[ x^2 + (484x^2 + 1100x + 625) + 13x + 66x + 75 = 0 \]
Combining like terms:
\[ 485x^2 + 1189x + 700 = 0 \]

### Step 4: Solve the quadratic equation for x
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ x = \frac{-1189 \pm \sqrt{1189^2 - 4 \cdot 485 \cdot 700}}{2 \cdot 485} \]
Calculating the discriminant:
\[ 1189^2 - 4 \cdot 485 \cdot 700 = 1415921 - 1366000 = 49921 \]
Now substituting back:
\[ x = \frac{-1189 \pm 223}{970} \]
Calculating the two possible values for \(x\):
1. \(x_1 = \frac{-966}{970} = -0.995\)
2. \(x_2 = \frac{-1412}{970} = -1.45\)

### Step 5: Find corresponding y values
Using \(y = -22x - 25\):
For \(x_1 = -0.995\):
\[ y_1 = -22(-0.995) - 25 = 21.89 - 25 = -3.11 \]
For \(x_2 = -1.45\):
\[ y_2 = -22(-1.45) - 25 = 31.9 - 25 = 6.9 \]

### Step 6: Write the equation of the circle through (1, 1) and the intersection points
The general form of the circle can be written as:
\[ x^2 + y^2 + Dx + Ey + F = 0 \]
Substituting the points (1, 1), (-0.995, -3.11), and (-1.45, 6.9) into this equation will yield a system of equations to solve for \(D\), \(E\), and \(F\).

### Final Step: Solve for D, E, F
Substituting the points into the circle equation will give three equations. Solving this system will provide the values for \(D\), \(E\), and \(F\).