If AFB is a focal chord of the parabola `y^(2) = 4ax` such that `AF = 4` and `FB = 5` then the latus-rectum of the parabola is equal to (a) 80 (b)`9/80` (c) 9 (d) `80/9`

Options

A. 80
B. `(9)/(80)`
C. 9
D. `(80)/(9)`

Explanation

To solve the problem, we need to find the latus rectum of the parabola given that AFB is a focal chord, with AF = 4 and FB = 5.

### Step-by-Step Solution:

1. **Identify the equation of the parabola**:
The equation of the parabola is given as \( y^2 = 4ax \).

2. **Understand the properties of focal chords**:
For a parabola, if AFB is a focal chord, the relationship between the segments of the focal chord and the parameter \( a \) is given by:
\[ \frac{1}{a} = \frac{1}{AF} + \frac{1}{FB} \]

3. **Substitute the values of AF and FB**:
We know that \( AF = 4 \) and \( FB = 5 \). Substituting these values into the equation gives:
\[ \frac{1}{a} = \frac{1}{4} + \frac{1}{5} \]

4. **Find a common denominator and add the fractions**:
The common denominator for 4 and 5 is 20. Thus:
\[ \frac{1}{4} = \frac{5}{20}, \quad \frac{1}{5} = \frac{4}{20} \]
Therefore:
\[ \frac{1}{a} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20} \]

5. **Calculate \( a \)**:
Taking the reciprocal gives:
\[ a = \frac{20}{9} \]

6. **Find the latus rectum**:
The length of the latus rectum \( L \) of the parabola is given by the formula:
\[ L = 4a \]
Substituting the value of \( a \):
\[ L = 4 \times \frac{20}{9} = \frac{80}{9} \]

7. **Conclusion**:
Therefore, the latus rectum of the parabola is \( \frac{80}{9} \).

### Final Answer:
The latus rectum of the parabola is \( \frac{80}{9} \).