Find the values of `a` and `b` if the slope of the tangent to the curve `x y+a x+b y=2` at `(1,\ 1)` is 2.

Explanation

To solve the problem, we need to find the values of `a` and `b` such that the slope of the tangent to the curve \( xy + ax + by = 2 \) at the point \( (1, 1) \) is equal to 2.

### Step-by-Step Solution:

1. **Differentiate the equation implicitly**:
We start with the equation of the curve:
\[ xy + ax + by = 2 \]
We differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(xy) + \frac{d}{dx}(ax) + \frac{d}{dx}(by) = \frac{d}{dx}(2) \]
Using the product rule on \( xy \):
\[ y + x \frac{dy}{dx} + a + b \frac{dy}{dx} = 0 \]

2. **Rearranging the equation**:
We can rearrange this to isolate \( \frac{dy}{dx} \):
\[ x \frac{dy}{dx} + b \frac{dy}{dx} = -y - a \]
Factoring out \( \frac{dy}{dx} \):
\[ (x + b) \frac{dy}{dx} = -y - a \]
Thus,
\[ \frac{dy}{dx} = \frac{-y - a}{x + b} \]

3. **Substituting the point (1, 1)**:
We substitute \( x = 1 \) and \( y = 1 \) into the derivative:
\[ \frac{dy}{dx} = \frac{-(1) - a}{1 + b} \]
We know that the slope at this point is 2, so we set up the equation:
\[ \frac{-(1 + a)}{1 + b} = 2 \]

4. **Cross-multiplying**:
Cross-multiplying gives:
\[ -(1 + a) = 2(1 + b) \]
Expanding both sides:
\[ -1 - a = 2 + 2b \]
Rearranging this gives us our first equation:
\[ a + 2b = -3 \quad \text{(Equation 1)} \]

5. **Substituting into the original equation**:
Now we substitute \( (1, 1) \) into the original equation to find another relationship:
\[ 1 \cdot 1 + a \cdot 1 + b \cdot 1 = 2 \]
This simplifies to:
\[ 1 + a + b = 2 \]
Rearranging gives us our second equation:
\[ a + b = 1 \quad \text{(Equation 2)} \]

6. **Solving the system of equations**:
We now have two equations:
- \( a + 2b = -3 \) (Equation 1)
- \( a + b = 1 \) (Equation 2)

We can subtract Equation 2 from Equation 1:
\[ (a + 2b) - (a + b) = -3 - 1 \]
This simplifies to:
\[ b = -4 \]

7. **Finding \( a \)**:
Substitute \( b = -4 \) back into Equation 2:
\[ a + (-4) = 1 \]
Thus,
\[ a = 5 \]

### Final Values:
The values of \( a \) and \( b \) are:
\[ a = 5, \quad b = -4 \]