If the line` x + y = 1 `touches the parabola `y^2-y + x = 0`, then the coordinates of the point of contact are:

Options

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

Answer

To solve the problem, we need to determine the coordinates of the point where the line \( x + y = 1 \) touches the parabola \( y^2 - y + x = 0 \).

### Step-by-Step Solution:

1. **Rewrite the Line Equation**:
The equation of the line is given as:
\[ x + y = 1 \]
We can express \( x \) in terms of \( y \):
\[ x = 1 - y \]

2. **Substitute into the Parabola Equation**:
Now, we substitute \( x = 1 - y \) into the parabola equation \( y^2 - y + x = 0 \):
\[ y^2 - y + (1 - y) = 0 \]
Simplifying this gives:
\[ y^2 - 2y + 1 = 0 \]

3. **Factor the Quadratic**:
The equation \( y^2 - 2y + 1 = 0 \) can be factored as:
\[ (y - 1)^2 = 0 \]
This implies:
\[ y - 1 = 0 \quad \Rightarrow \quad y = 1 \]

4. **Find Corresponding x-value**:
Now that we have \( y = 1 \), we can find the corresponding \( x \) value using the line equation:
\[ x + 1 = 1 \quad \Rightarrow \quad x = 0 \]

5. **Coordinates of the Point of Contact**:
Therefore, the coordinates of the point of contact are:
\[ (x, y) = (0, 1) \]

### Final Answer:
The coordinates of the point of contact are \( (0, 1) \).

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