The curve given by `x+y=e^(x y)` has a tangent parallel to the y-axis at the point

Options

A. `(0,1)`
B. `(1,0)`
C. `(1,1)`
D. none of these

Answer

To find the point on the curve \( x + y = e^{xy} \) where the tangent is parallel to the y-axis, we need to follow these steps:

### Step 1: Differentiate the given equation
We start with the equation of the curve:
\[ x + y = e^{xy} \]
To find the slope of the tangent line, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(x) + \frac{d}{dx}(y) = \frac{d}{dx}(e^{xy}) \]
This gives us:
\[ 1 + \frac{dy}{dx} = e^{xy} \left( y + x \frac{dy}{dx} \right) \]

### Step 2: Rearrange the equation
We can rearrange the equation to isolate \( \frac{dy}{dx} \):
\[ 1 + \frac{dy}{dx} = e^{xy}y + e^{xy}x \frac{dy}{dx} \]
Rearranging gives:
\[ \frac{dy}{dx} - e^{xy}x \frac{dy}{dx} = e^{xy}y - 1 \]
Factoring out \( \frac{dy}{dx} \):
\[ \frac{dy}{dx}(1 - e^{xy}x) = e^{xy}y - 1 \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{e^{xy}y - 1}{1 - e^{xy}x} \]

### Step 3: Set the slope to be undefined
A tangent that is parallel to the y-axis means that the slope \( \frac{dy}{dx} \) is undefined. This occurs when the denominator is zero:
\[ 1 - e^{xy}x = 0 \]
This leads to:
\[ e^{xy}x = 1 \quad \Rightarrow \quad e^{xy} = \frac{1}{x} \]

### Step 4: Substitute back into the original equation
Now, we substitute \( e^{xy} = \frac{1}{x} \) back into the original equation:
\[ x + y = \frac{1}{x} \]
Rearranging gives:
\[ y = \frac{1}{x} - x \]

### Step 5: Solve for \( y \)
To find the specific points, we can substitute values for \( x \) and solve for \( y \). We also know that \( e^{xy} = \frac{1}{x} \) must hold true.

### Step 6: Check the options
We will check the options provided (0, 1), (0.1, 0), (1, 0), etc. to find which one satisfies both the original equation and the condition \( e^{xy} = \frac{1}{x} \).

1. For \( (0, 1) \):
- \( e^{0 \cdot 1} = 1 \) and \( \frac{1}{0} \) is undefined. Not valid.

2. For \( (0.1, 0) \):
- \( e^{0.1 \cdot 0} = 1 \) and \( \frac{1}{0.1} = 10 \). Not valid.

3. For \( (1, 0) \):
- \( e^{1 \cdot 0} = 1 \) and \( \frac{1}{1} = 1 \). Valid.

4. For \( (0.1, 1) \):
- \( e^{0.1 \cdot 1} = e^{0.1} \) which is not equal to \( 10 \). Not valid.

### Conclusion
The point on the curve where the tangent is parallel to the y-axis is:
\[ \boxed{(1, 0)} \]