Equation of the circle cutting orthogonal these circles `x^2+y^2-2x-3y-7=0`,`x^2 +y^2+5x-5y+9=0` and `x^2+y^2+7x-9y+29=0` is:

Answer

To find the equation of the circle that cuts orthogonally the given circles, we will follow these steps:

### Step 1: Write down the equations of the given circles
The equations of the circles are:
1. \( C_1: x^2 + y^2 - 2x - 3y - 7 = 0 \)
2. \( C_2: x^2 + y^2 + 5x - 5y + 9 = 0 \)
3. \( C_3: x^2 + y^2 + 7x - 9y + 29 = 0 \)

### Step 2: Find the radial axis of circles \( C_1 \) and \( C_2 \)
To find the radial axis, we subtract the equations of the circles:
\[ C_1 - C_2 = 0 \]
This gives us:
\[ (x^2 + y^2 - 2x - 3y - 7) - (x^2 + y^2 + 5x - 5y + 9) = 0 \]
Simplifying this, we get:
\[ -2x - 3y - 7 - 5x + 5y + 9 = 0 \]
Combining like terms:
\[ -7x + 2y + 2 = 0 \quad \text{(Equation 4)} \]
Rearranging gives:
\[ 7x - 2y - 2 = 0 \]

### Step 3: Find the radial axis of circles \( C_2 \) and \( C_3 \)
Now we find the radial axis between \( C_2 \) and \( C_3 \):
\[ C_2 - C_3 = 0 \]
This gives us:
\[ (x^2 + y^2 + 5x - 5y + 9) - (x^2 + y^2 + 7x - 9y + 29) = 0 \]
Simplifying this, we get:
\[ 5x - 5y + 9 - 7x + 9y - 29 = 0 \]
Combining like terms:
\[ -2x + 4y - 20 = 0 \quad \text{(Equation 5)} \]
Rearranging gives:
\[ 2x - 4y + 20 = 0 \]

### Step 4: Solve the equations (4) and (5) simultaneously
We have the two equations:
1. \( 7x - 2y - 2 = 0 \)
2. \( 2x - 4y + 20 = 0 \)

From the second equation, we can express \( y \) in terms of \( x \):
\[ 2x + 20 = 4y \implies y = \frac{1}{2}x + 5 \]

Substituting this into the first equation:
\[ 7x - 2\left(\frac{1}{2}x + 5\right) - 2 = 0 \]
Simplifying:
\[ 7x - x - 10 - 2 = 0 \implies 6x - 12 = 0 \implies x = 2 \]

Now substituting \( x = 2 \) back to find \( y \):
\[ y = \frac{1}{2}(2) + 5 = 1 + 5 = 6 \]
Thus, the center of the circle is \( (2, 6) \).

### Step 5: Find the radius of the circle
Using the formula for the radius:
\[ r = \sqrt{g^2 + f^2 - c} \]
From the equation of the circle, we can find \( g, f, c \):
For the center \( (h, k) = (2, 6) \), we need to find \( g, f, c \) from one of the original circles. We can use \( C_1 \):
\[ g = -2, f = -3, c = -7 \]
Calculating:
\[ r = \sqrt{(-2)^2 + (-3)^2 - (-7)} = \sqrt{4 + 9 + 7} = \sqrt{20} = 2\sqrt{5} \]

### Step 6: Write the equation of the circle
The equation of the circle is:
\[ (x - 2)^2 + (y - 6)^2 = (2\sqrt{5})^2 \]
Expanding this gives:
\[ (x - 2)^2 + (y - 6)^2 = 20 \]
Expanding further:
\[ x^2 - 4x + 4 + y^2 - 12y + 36 = 20 \]
Combining terms:
\[ x^2 + y^2 - 4x - 12y + 20 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 4x - 12y + 20 = 0 \]