The curve `y-e^(xy)+x=0` has a vertical tangent at the point :

Options

A. (1, 1)
B. at no point
C. (0, 1)
D. (1, 0)

Explanation

To find the point where the curve \( y - e^{xy} + x = 0 \) has a vertical tangent, we need to determine where the derivative \( \frac{dy}{dx} \) is undefined, which typically occurs when the denominator of the derivative equals zero.

### Step-by-Step Solution:

1. **Differentiate the given equation**:
We start with the equation of the curve:
\[ y - e^{xy} + x = 0 \]
We will differentiate both sides with respect to \( x \).

2. **Apply implicit differentiation**:
Differentiating \( y \) gives \( \frac{dy}{dx} \). For \( e^{xy} \), we use the chain rule:
\[ \frac{d}{dx}(e^{xy}) = e^{xy} \cdot \left( y + x \frac{dy}{dx} \right) \]
The derivative of \( x \) is simply \( 1 \). Thus, differentiating the entire equation gives:
\[ \frac{dy}{dx} - e^{xy} \left( y + x \frac{dy}{dx} \right) + 1 = 0 \]

3. **Rearranging the equation**:
Rearranging the equation to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} - e^{xy} y - x e^{xy} \frac{dy}{dx} + 1 = 0 \]
Combine the \( \frac{dy}{dx} \) terms:
\[ \frac{dy}{dx} (1 - x e^{xy}) = e^{xy} y - 1 \]

4. **Solve for \( \frac{dy}{dx} \)**:
Now, we can express \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{e^{xy} y - 1}{1 - x e^{xy}} \]

5. **Determine where the tangent is vertical**:
A vertical tangent occurs when the denominator is zero:
\[ 1 - x e^{xy} = 0 \implies x e^{xy} = 1 \]

6. **Substituting points**:
We need to find the points \( (x, y) \) that satisfy this equation. We will check the provided options to see which one satisfies this condition.

- For point A: \( (1, 1) \)
\[ 1 \cdot e^{1 \cdot 1} = e \quad (\text{not } 1) \]
- For point B: \( (0, 0) \)
\[ 0 \cdot e^{0 \cdot 0} = 0 \quad (\text{not } 1) \]
- For point C: \( (1, 0) \)
\[ 1 \cdot e^{1 \cdot 0} = 1 \quad (\text{satisfies}) \]
- For point D: \( (1, 0) \)
\[ 1 \cdot e^{1 \cdot 0} = 1 \quad (\text{satisfies}) \]

7. **Conclusion**:
The point where the curve has a vertical tangent is:
\[ (1, 0) \]