If the line `x+y=a` touches the parabola `y=x-x^2,` then find the value of `adot`

Explanation

To solve the problem, we need to find the value of \( a \) such that the line \( x + y = a \) touches the parabola \( y = x - x^2 \).

### Step-by-Step Solution:

1. **Rewrite the line equation**:
The line can be rewritten as:
\[ y = a - x \]

2. **Substitute the line equation into the parabola equation**:
We substitute \( y \) from the line equation into the parabola equation:
\[ a - x = x - x^2 \]

3. **Rearrange the equation**:
Bringing all terms to one side gives:
\[ x^2 - 2x + a = 0 \]

4. **Condition for tangency**:
For the line to touch the parabola, the quadratic equation must have exactly one solution. This occurs when the discriminant is zero. The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
In our case, \( A = 1 \), \( B = -2 \), and \( C = a \).

5. **Set the discriminant to zero**:
We set the discriminant equal to zero:
\[ (-2)^2 - 4(1)(a) = 0 \]
Simplifying this gives:
\[ 4 - 4a = 0 \]

6. **Solve for \( a \)**:
Rearranging the equation gives:
\[ 4 = 4a \]
Dividing both sides by 4:
\[ a = 1 \]

### Final Answer:
The value of \( a \) is \( 1 \).

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