The diameter of the parabola `y^2=6x` corresponding to the system of parallel chords 3x-y+c=0 is

Options

A. (a)y-1=0
B. (b)y-2=0
C. (c)y+1=0
D. (d)y+2=0

Answer

To solve the problem of finding the diameter of the parabola \( y^2 = 6x \) corresponding to the system of parallel chords given by the equation \( 3x - y + c = 0 \), we can follow these steps:

### Step 1: Identify the parabola and its properties
The given parabola is \( y^2 = 6x \). This is a standard form of a parabola that opens to the right. The vertex of this parabola is at the origin (0, 0).

### Step 2: Determine the slope of the chords
The equation of the parallel chords is given as \( 3x - y + c = 0 \). We can rearrange this to find the slope:
\[ y = 3x + c \]
The slope of these lines is 3.

### Step 3: Find the midpoint of the chords
Let \( (h, k) \) be the midpoint of the chord of the parabola. The equation of the chord of contact for a point \( (h, k) \) on the parabola is given by:
\[ T = S_1 \]
where \( T \) is the equation of the chord of contact, and \( S_1 \) is the equation of the parabola evaluated at the point \( (h, k) \).

For the parabola \( y^2 = 6x \), we have:
\[ S_1 = k^2 = 6h \]

### Step 4: Write the equation of the chord of contact
The chord of contact can also be expressed as:
\[ k y = 3x + h \]
Substituting \( k = 3 \) (the slope of the chords) into the equation gives:
\[ y = 3x + h \]

### Step 5: Equate the slopes
Since the slopes of the chord of contact and the given lines must be equal, we have:
\[ \frac{3}{k} = -\frac{1}{3} \]
This gives:
\[ k = 1 \]

### Step 6: Find the locus of the midpoint
Now, substituting \( k = 1 \) into the equation of the chord gives us:
\[ y - k = 0 \implies y - 1 = 0 \]
This means the locus of the midpoints of the chords is the line \( y = 1 \).

### Step 7: Determine the diameter of the parabola
The diameter of the parabola corresponds to the distance between the points where the line \( y = 1 \) intersects the parabola. To find these points, we substitute \( y = 1 \) into the parabola's equation:
\[ 1^2 = 6x \implies 1 = 6x \implies x = \frac{1}{6} \]
Thus, the points of intersection are \( \left( \frac{1}{6}, 1 \right) \) and the diameter is the distance across the parabola at this line.

### Step 8: Conclusion
The diameter of the parabola corresponding to the line \( y = 1 \) is simply the horizontal distance across the parabola at that height, which is determined by the points of intersection.

### Final Answer
The diameter of the parabola corresponding to the system of parallel chords is \( 1 \).

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