If a rectangular hyperbola `(x-1)(y-2)=4` cuts a circle `x^(2)+y^(2)+2gx+2fy+c=0` at points `(3, 4), (5, 3), (2, 6) and (-1, 0)`, then the value of (g+f) is equal to

Options

A. `-3`
B. `-9`
C. `8`
D. `9`

Answer

To solve the problem step by step, we need to find the value of \( g + f \) given that the rectangular hyperbola \((x-1)(y-2)=4\) intersects the circle \(x^2 + y^2 + 2gx + 2fy + c = 0\) at the points \((3, 4)\), \((5, 3)\), \((2, 6)\), and \((-1, 0)\).

### Step 1: Substitute the points into the circle equation
We will substitute each of the given points into the circle equation to form equations involving \(g\), \(f\), and \(c\).

1. **For the point \((3, 4)\)**:
\[ 3^2 + 4^2 + 2g(3) + 2f(4) + c = 0 \]
\[ 9 + 16 + 6g + 8f + c = 0 \implies 25 + 6g + 8f + c = 0 \implies 6g + 8f + c = -25 \quad \text{(Equation 1)} \]

2. **For the point \((5, 3)\)**:
\[ 5^2 + 3^2 + 2g(5) + 2f(3) + c = 0 \]
\[ 25 + 9 + 10g + 6f + c = 0 \implies 34 + 10g + 6f + c = 0 \implies 10g + 6f + c = -34 \quad \text{(Equation 2)} \]

3. **For the point \((2, 6)\)**:
\[ 2^2 + 6^2 + 2g(2) + 2f(6) + c = 0 \]
\[ 4 + 36 + 4g + 12f + c = 0 \implies 40 + 4g + 12f + c = 0 \implies 4g + 12f + c = -40 \quad \text{(Equation 3)} \]

4. **For the point \((-1, 0)\)**:
\[ (-1)^2 + 0^2 + 2g(-1) + 2f(0) + c = 0 \]
\[ 1 + 0 - 2g + c = 0 \implies -2g + c = -1 \implies c = 2g - 1 \quad \text{(Equation 4)} \]

### Step 2: Substitute \(c\) from Equation 4 into the other equations
Now we will substitute \(c = 2g - 1\) into Equations 1, 2, and 3.

1. **Substituting into Equation 1**:
\[ 6g + 8f + (2g - 1) = -25 \]
\[ 8g + 8f - 1 = -25 \implies 8g + 8f = -24 \implies g + f = -3 \quad \text{(Equation 5)} \]

2. **Substituting into Equation 2**:
\[ 10g + 6f + (2g - 1) = -34 \]
\[ 12g + 6f - 1 = -34 \implies 12g + 6f = -33 \implies 2g + f = -5.5 \quad \text{(Equation 6)} \]

3. **Substituting into Equation 3**:
\[ 4g + 12f + (2g - 1) = -40 \]
\[ 6g + 12f - 1 = -40 \implies 6g + 12f = -39 \implies g + 2f = -6.5 \quad \text{(Equation 7)} \]

### Step 3: Solve for \(g + f\)
From Equation 5, we already have:
\[ g + f = -3 \]

### Conclusion
Thus, the value of \(g + f\) is \(-3\).