A curve is represented parametrically by the equations `x=t+e^(at) and y=-t+e^(at)` when `t in R and a > 0.` If the curve touches the axis of x at the point A, then the coordinates of the point A are

Options

A. `(1,0)`
B. `(2e,0)`
C. `(e,0)`
D. `(1//e,0)`

Explanation

To find the coordinates of the point A where the curve touches the x-axis, we start with the given parametric equations:

1. **Equations:**
\[ x = t + e^{at} \]
\[ y = -t + e^{at} \]

2. **Condition for touching the x-axis:**
The curve touches the x-axis at point A, which means at this point \( y = 0 \) and the slope of the curve \( \frac{dy}{dx} = 0 \).

3. **Finding \( t_1 \) where \( y = 0 \):**
Set \( y = 0 \):
\[ -t_1 + e^{at_1} = 0 \]
Rearranging gives:
\[ e^{at_1} = t_1 \]
This is our first equation.

4. **Finding the derivative \( \frac{dy}{dx} \):**
We need to find \( \frac{dy}{dx} \) using the chain rule:
\[ \frac{dy}{dt} = -1 + ae^{at} \]
\[ \frac{dx}{dt} = 1 + ae^{at} \]
Thus,
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-1 + ae^{at}}{1 + ae^{at}} \]

5. **Setting the derivative to zero:**
For the curve to touch the x-axis, we set \( \frac{dy}{dx} = 0 \):
\[ -1 + ae^{at_1} = 0 \]
Rearranging gives:
\[ ae^{at_1} = 1 \]
This is our second equation.

6. **Solving the equations:**
Now we have two equations:
1. \( e^{at_1} = t_1 \)
2. \( ae^{at_1} = 1 \)

From the second equation, we can express \( a \):
\[ a = \frac{1}{e^{at_1}} = \frac{1}{t_1} \]

7. **Substituting \( a \) into the first equation:**
Substitute \( a \) into \( e^{at_1} = t_1 \):
\[ e^{\frac{t_1}{t_1}} = t_1 \implies e = t_1 \]
Thus, we find:
\[ t_1 = e \]

8. **Finding \( a \):**
Now substituting \( t_1 = e \) back into \( a = \frac{1}{t_1} \):
\[ a = \frac{1}{e} \]

9. **Finding coordinates of point A:**
Now we can find the coordinates of point A:
\[ x = t_1 + e^{at_1} = e + e^{\frac{1}{e} \cdot e} = e + e = 2e \]
\[ y = 0 \]
Therefore, the coordinates of point A are:
\[ A(2e, 0) \]

### Summary of the Solution:
The coordinates of the point A where the curve touches the x-axis are \( (2e, 0) \).