Find the equation of the circle which cuts orthogonally the circle `x^2+y^2-4x+2y-7=0` and having the centre at (2,3)

Answer

To find the equation of the circle that cuts orthogonally with the given circle and has its center at (2, 3), we will follow these steps:

### Step 1: Identify the given circle's equation
The equation of the given circle is:
\[ x^2 + y^2 - 4x + 2y - 7 = 0 \]

### Step 2: Rewrite the equation in standard form
To find the center and radius of the given circle, we will complete the square for both \(x\) and \(y\).

1. Rearranging the equation:
\[ x^2 - 4x + y^2 + 2y = 7 \]

2. Completing the square for \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

3. Completing the square for \(y\):
\[ y^2 + 2y = (y + 1)^2 - 1 \]

4. Substitute back into the equation:
\[ (x - 2)^2 - 4 + (y + 1)^2 - 1 = 7 \]
\[ (x - 2)^2 + (y + 1)^2 = 12 \]

Thus, the center \(C_1\) of the given circle is \((2, -1)\) and the radius \(r_1 = \sqrt{12}\).

### Step 3: Identify the center of the required circle
The center of the required circle is given as \(C_2(2, 3)\).

### Step 4: Use the orthogonality condition
Two circles cut orthogonally if the following condition holds:
\[ 2g \cdot g' + 2f \cdot f' = c + c' \]
where \(g, f, c\) are the coefficients of the required circle and \(g', f', c'\) are the coefficients of the given circle.

From the standard form of the circle:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

### Step 5: Assign values based on the center
Since the center of the required circle is \((2, 3)\):
- \(g = -2\) (because the center is \(-g\))
- \(f = -3\) (because the center is \(-f\))

### Step 6: Find the value of \(c\)
From the given circle, we have:
- \(g' = -2\)
- \(f' = 1\)
- \(c' = -7\)

Substituting these values into the orthogonality condition:
\[ 2(-2)(-2) + 2(-3)(1) = c - 7 \]
Calculating:
\[ 8 - 6 = c - 7 \]
\[ 2 = c - 7 \]
\[ c = 9 \]

### Step 7: Write the equation of the required circle
Now substituting \(g\), \(f\), and \(c\) into the standard form:
\[ x^2 + y^2 - 4x - 6y + 9 = 0 \]

### Final Equation
Thus, the equation of the required circle is:
\[ x^2 + y^2 - 4x - 6y + 9 = 0 \]

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