The equation of the chord of the parabola `y^(2)=2x` having (1,1) as its midpoint is

Options

A. x+y=0
B. x-y=0
C. x-y+1=0
D. 2x-y=0

Answer

To find the equation of the chord of the parabola \( y^2 = 2x \) that has the point (1, 1) as its midpoint, we can use the formula for the equation of a chord with a given midpoint.

### Step-by-Step Solution:

1. **Identify the Parabola and its Parameters**:
The given parabola is \( y^2 = 2x \). Here, we can identify \( 4a = 2 \), which gives \( a = \frac{1}{2} \).

2. **Use the Chord Midpoint Formula**:
The formula for the equation of a chord of the parabola \( y^2 = 4ax \) with midpoint \( (x_1, y_1) \) is:
\[ yy_1 = 2a(x + x_1) \]
In our case, the midpoint is \( (1, 1) \), so \( x_1 = 1 \) and \( y_1 = 1 \).

3. **Substitute the Values into the Formula**:
Substituting \( a = \frac{1}{2} \), \( x_1 = 1 \), and \( y_1 = 1 \) into the chord formula:
\[ y \cdot 1 = 2 \cdot \frac{1}{2}(x + 1) \]
This simplifies to:
\[ y = (x + 1) \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ y - x - 1 = 0 \]
This is the equation of the chord.

5. **Final Answer**:
Thus, the equation of the chord of the parabola \( y^2 = 2x \) having (1, 1) as its midpoint is:
\[ y - x - 1 = 0 \]