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. a) `-3`
B. b) `-9`
C. c) `8`
D. d) `9`

Answer

To find the value of \( g + f \) given the hyperbola and the circle, we can follow these steps:

### Step 1: Identify the points of intersection
The points where the hyperbola \( (x-1)(y-2) = 4 \) intersects the circle are given as \( (3, 4), (5, 3), (2, 6), (-1, 0) \).

### Step 2: Use the formula for the sum of x-coordinates
We will use the formula:
\[ \frac{\sum x_i}{4} = -\frac{g}{2} + \frac{1}{2} \]
where \( x_i \) are the x-coordinates of the intersection points.

### Step 3: Calculate the sum of x-coordinates
Calculating the sum of the x-coordinates:
\[ 3 + 5 + 2 - 1 = 9 \]
Now substituting into the formula:
\[ \frac{9}{4} = -\frac{g}{2} + \frac{1}{2} \]

### Step 4: Rearranging the equation for g
Rearranging gives:
\[ -\frac{g}{2} = \frac{9}{4} - \frac{1}{2} \]
Converting \( \frac{1}{2} \) to a fraction with a denominator of 4:
\[ -\frac{g}{2} = \frac{9}{4} - \frac{2}{4} = \frac{7}{4} \]
Multiplying both sides by -2:
\[ g = -\frac{7}{2} \]

### Step 5: Use the formula for the sum of y-coordinates
Now we will use the formula:
\[ \frac{\sum y_i}{4} = -\frac{f}{2} + \frac{1}{2} \]
Calculating the sum of the y-coordinates:
\[ 4 + 3 + 6 + 0 = 13 \]
Substituting into the formula:
\[ \frac{13}{4} = -\frac{f}{2} + \frac{1}{2} \]

### Step 6: Rearranging the equation for f
Rearranging gives:
\[ -\frac{f}{2} = \frac{13}{4} - \frac{1}{2} \]
Converting \( \frac{1}{2} \) to a fraction with a denominator of 4:
\[ -\frac{f}{2} = \frac{13}{4} - \frac{2}{4} = \frac{11}{4} \]
Multiplying both sides by -2:
\[ f = -\frac{11}{2} \]

### Step 7: Calculate g + f
Now we can find \( g + f \):
\[ g + f = -\frac{7}{2} - \frac{11}{2} = -\frac{18}{2} = -9 \]

### Final Answer
Thus, the value of \( g + f \) is:
\[ \boxed{-9} \]