The number of focal chord(s) of length 4/7 in the parabola `7y^(2)=8x` is

Options

A. 1
B. zero
C. infinite
D. none of these

Answer

To solve the problem of finding the number of focal chords of length \( \frac{4}{7} \) in the parabola given by the equation \( 7y^2 = 8x \), we can follow these steps:

### Step 1: Rewrite the equation of the parabola
The given equation of the parabola is:
\[ 7y^2 = 8x \]
We can rewrite it in the standard form \( y^2 = 4ax \):
\[ y^2 = \frac{8}{7}x \]
From this, we can identify \( 4a = \frac{8}{7} \).

### Step 2: Find the value of \( a \)
To find \( a \), we divide both sides by 4:
\[ a = \frac{8}{7 \times 4} = \frac{2}{7} \]

### Step 3: Determine the minimum length of a focal chord
The minimum length of a focal chord for a parabola is given by the formula \( 4a \):
\[ \text{Minimum length of focal chord} = 4a = 4 \times \frac{2}{7} = \frac{8}{7} \]

### Step 4: Compare the given length with the minimum length
The problem states that we are looking for focal chords of length \( \frac{4}{7} \). We need to compare this with the minimum length of the focal chord:
\[ \frac{4}{7} < \frac{8}{7} \]
Since \( \frac{4}{7} \) is less than \( \frac{8}{7} \), it is less than the minimum length of the focal chord.

### Step 5: Conclusion
Since the length of the focal chord must be greater than or equal to \( 4a \) and the given length \( \frac{4}{7} \) is less than this minimum length, we conclude that there are no focal chords of this length in the given parabola.

Thus, the number of focal chords of length \( \frac{4}{7} \) in the parabola \( 7y^2 = 8x \) is:
\[ \boxed{0} \]