If the line `y=x-1` bisects two chords of the parabola `y^(2)=4bx` which are passing through the point `(b, -2b)`, then the length of the latus rectum can be equal to

Options

A. 3
B. 5
C. 6
D. 8

Answer

To solve the problem, we need to find the length of the latus rectum of the parabola given that the line \( y = x - 1 \) bisects two chords of the parabola \( y^2 = 4bx \) that pass through the point \( (b, -2b) \).

### Step-by-Step Solution:

1. **Identify the parabola and its properties:**
The equation of the parabola is given by \( y^2 = 4bx \). The length of the latus rectum of a parabola of the form \( y^2 = 4px \) is \( 4p \). Here, \( p = b \), so the length of the latus rectum is \( 4b \).

2. **Find the coordinates of the point through which the chords pass:**
The chords pass through the point \( (b, -2b) \).

3. **Substitute the point into the parabola's equation:**
Substitute \( x = b \) and \( y = -2b \) into the parabola's equation:
\[ (-2b)^2 = 4b(b) \]
Simplifying gives:
\[ 4b^2 = 4b^2 \]
This confirms that the point \( (b, -2b) \) lies on the parabola.

4. **Find the intersection of the line with the parabola:**
The line \( y = x - 1 \) intersects the parabola. Substitute \( y \) in the parabola's equation:
\[ (x - 1)^2 = 4bx \]
Expanding this gives:
\[ x^2 - 2x + 1 = 4bx \]
Rearranging results in:
\[ x^2 - (4b + 2)x + 1 = 0 \]

5. **Use the property of the bisector:**
Since the line bisects the chords, the sum of the roots of the quadratic equation (which represent the x-coordinates of the intersection points) is equal to the x-coordinate of the midpoint of the chord. The sum of the roots \( S \) is given by:
\[ S = 4b + 2 \]
The midpoint of the chord can be calculated using the line \( y = x - 1 \). The midpoint's x-coordinate is given by \( h \) and the y-coordinate is \( h - 1 \).

6. **Set up the equation for the midpoint:**
The midpoint of the chord can also be expressed as:
\[ \frac{x_1 + x_2}{2} = h \]
Thus, \( x_1 + x_2 = 2h \). Equating gives:
\[ 2h = 4b + 2 \implies h = 2b + 1 \]

7. **Substitute back into the parabola's equation:**
Substitute \( h \) back into the parabola's equation to find the condition on \( b \):
\[ (2b + 1 - 1)^2 = 4b(2b + 1) \]
Simplifying gives:
\[ (2b)^2 = 8b^2 + 4b \implies 4b^2 = 8b^2 + 4b \]
Rearranging gives:
\[ 4b^2 + 4b = 0 \implies 4b(b + 1) = 0 \]
Thus, \( b = 0 \) or \( b = -1 \).

8. **Determine the length of the latus rectum:**
Since \( b \) must be positive for the parabola to be defined, we only consider \( b = 1 \). Therefore, the length of the latus rectum is:
\[ 4b = 4 \times 1 = 4 \]

### Conclusion:
The length of the latus rectum can be equal to **4**.