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 will follow these steps:

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

### Step 2: Determine the Value of \( a \)
From the equation \( 4a = \frac{8}{7} \), we can solve for \( a \):
\[ a = \frac{8}{28} = \frac{2}{7} \]

### Step 3: Calculate the Length of the Latus Rectum
The length of the latus rectum (which is the minimum length of a focal chord) is given by the formula \( 4a \):
\[ \text{Length of Latus Rectum} = 4a = 4 \times \frac{2}{7} = \frac{8}{7} \]

### Step 4: Compare the Given Length of Focal Chord with Latus Rectum
We are given that the length of the focal chord is \( \frac{4}{7} \). Now we need to compare this with the length of the latus rectum:
\[ \frac{4}{7} < \frac{8}{7} \]
Since the length of the focal chord \( \frac{4}{7} \) is less than the minimum length of the latus rectum \( \frac{8}{7} \), it implies that no focal chords of this length can exist.

### Step 5: Conclusion
Thus, the number of focal chords of length \( \frac{4}{7} \) in the parabola \( 7y^2 = 8x \) is:
\[ \boxed{0} \]