Point of contact of y=1-x w.r.t. `y^(2)-y+x=0` is

Options

A. a) (1,1)
B. b) `((1)/(2),(1)/(2))`
C. c) (0,1)
D. d) (1,0)

Answer

To find the point of contact of the line \( y = 1 - x \) with respect to the parabola \( y^2 - y + x = 0 \), we can follow these steps:

### Step 1: Substitute the line equation into the parabola equation
We start with the equations:
1. \( y = 1 - x \) (Equation 1)
2. \( y^2 - y + x = 0 \) (Equation 2)

Since we are looking for the point of contact, we will substitute Equation 1 into Equation 2. First, we express \( x \) in terms of \( y \):
\[ x = 1 - y \]

### Step 2: Substitute \( x \) into the parabola equation
Now, we substitute \( x = 1 - y \) into Equation 2:
\[ y^2 - y + (1 - y) = 0 \]
This simplifies to:
\[ y^2 - 2y + 1 = 0 \]

### Step 3: Factor or solve the quadratic equation
The equation \( y^2 - 2y + 1 = 0 \) can be factored as:
\[ (y - 1)^2 = 0 \]
This gives us:
\[ y - 1 = 0 \implies y = 1 \]

### Step 4: Find the corresponding \( x \) value
Now that we have \( y = 1 \), we can find the corresponding \( x \) value using Equation 1:
\[ x = 1 - y = 1 - 1 = 0 \]

### Step 5: Write the point of contact
The point of contact is therefore:
\[ (x, y) = (0, 1) \]

### Final Answer
The point of contact of the line \( y = 1 - x \) with respect to the parabola \( y^2 - y + x = 0 \) is \( (0, 1) \).

---