Length of the shortest chord of the parabola `y^2=4x+8`, which belongs to the family of lines `(1+lambda)y+(lambda-1)x+2(1-lambda)=0` is

Options

A. 6
B. 5
C. 8
D. 2

Answer

To find the length of the shortest chord of the parabola \( y^2 = 4x + 8 \) that belongs to the family of lines given by \( (1+\lambda)y + (\lambda - 1)x + 2(1 - \lambda) = 0 \), we can follow these steps:

### Step 1: Rewrite the equation of the line
We start by rearranging the line equation:

\[ (1+\lambda)y + (\lambda - 1)x + 2(1 - \lambda) = 0 \]

Distributing \( y \) and \( x \):

\[ y + \lambda y + \lambda x - x + 2 - 2\lambda = 0 \]

This simplifies to:

\[ y - x + 2 + \lambda(y + x - 2) = 0 \]

### Step 2: Identify the lines in the family
This equation can be expressed in the form \( L_1 + \lambda L_2 = 0 \), where:

- \( L_1: y - x + 2 = 0 \) (Line 1)
- \( L_2: y + x - 2 = 0 \) (Line 2)

### Step 3: Find the intersection of the lines
To find the intersection of these two lines, we solve them simultaneously:

1. From \( L_1: y - x + 2 = 0 \), we can express \( y \) as:
\[ y = x - 2 \]

2. Substitute this into \( L_2: y + x - 2 = 0 \):
\[ (x - 2) + x - 2 = 0 \]
\[ 2x - 4 = 0 \implies x = 2 \]

3. Substitute \( x = 2 \) back into \( y = x - 2 \):
\[ y = 2 - 2 = 0 \]

Thus, the point of intersection is \( (2, 0) \).

### Step 4: Analyze the parabola
The equation of the parabola is:

\[ y^2 = 4x + 8 \]

We can rewrite it as:

\[ y^2 = 4(x + 2) \]

This indicates that the parabola opens to the right, with its vertex at \( (-2, 0) \).

### Step 5: Determine the shortest chord
The shortest chord of a parabola that is perpendicular to the axis of symmetry is a double ordinate. Since the x-coordinate of the intersection point is \( 2 \), we consider the vertical line \( x = 2 \).

### Step 6: Find the y-coordinates of the endpoints of the chord
Substituting \( x = 2 \) into the parabola's equation to find the y-coordinates:

\[ y^2 = 4(2) + 8 = 8 + 8 = 16 \implies y = \pm 4 \]

Thus, the points on the parabola corresponding to \( x = 2 \) are \( (2, 4) \) and \( (2, -4) \).

### Step 7: Calculate the length of the chord
The length of the chord \( AB \) is the distance between the points \( (2, 4) \) and \( (2, -4) \):

\[ \text{Length of chord} = |4 - (-4)| = |4 + 4| = 8 \]

### Final Answer
The length of the shortest chord of the parabola is \( 8 \).

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