Let the focus S of the parabola `y^(2)=8x` lie on the focal chord PQ of the same parabola. If the length QS = 3 units, then the ratio of length PQ to the length of the laturs rectum of the parabola is

Options

A. `(2)/(sqrt5)`
B. `(4)/(5)`
C. `(5)/(4)`
D. `(9)/(8)`

Answer

To solve the problem, we need to find the ratio of the length of the focal chord \( PQ \) to the length of the latus rectum of the parabola given by the equation \( y^2 = 8x \).

### Step-by-Step Solution:

1. **Identify the parameters of the parabola**:
The given parabola is \( y^2 = 8x \).
- The standard form of a parabola is \( y^2 = 4ax \).
- Here, \( 4a = 8 \) implies \( a = 2 \).

2. **Locate the focus of the parabola**:
The focus \( S \) of the parabola \( y^2 = 8x \) is at the point \( (a, 0) = (2, 0) \).

3. **Understand the focal chord**:
A focal chord is a line segment that passes through the focus and has its endpoints on the parabola. Let the endpoints of the focal chord be \( P \) and \( Q \).

4. **Use the property of harmonic progression**:
The distances \( PS \), \( SQ \), and the distance from the focus to the directrix (which is \( 4 \) for this parabola) are in harmonic progression.
- Let \( PS = p \) and \( SQ = 3 \) (as given in the problem).
- The distance from the focus to the directrix is \( 4 \).

5. **Set up the harmonic progression relationship**:
Since \( PS \), \( SQ \), and \( 4 \) are in harmonic progression, we have:
\[ \frac{1}{PS} + \frac{1}{SQ} = \frac{2}{4} \]
Substituting \( SQ = 3 \):
\[ \frac{1}{p} + \frac{1}{3} = \frac{1}{2} \]

6. **Solve for \( PS \)**:
Rearranging the equation:
\[ \frac{1}{p} = \frac{1}{2} - \frac{1}{3} \]
Finding a common denominator (which is 6):
\[ \frac{1}{p} = \frac{3 - 2}{6} = \frac{1}{6} \]
Thus, \( p = 6 \).

7. **Calculate the length of the focal chord \( PQ \)**:
The total length of the focal chord \( PQ \) is:
\[ PQ = PS + SQ = 6 + 3 = 9 \]

8. **Find the length of the latus rectum**:
The length of the latus rectum for the parabola \( y^2 = 4ax \) is given by \( 4a \):
\[ \text{Length of latus rectum} = 4 \times 2 = 8 \]

9. **Calculate the ratio of \( PQ \) to the length of the latus rectum**:
\[ \text{Ratio} = \frac{PQ}{\text{Length of latus rectum}} = \frac{9}{8} \]

### Final Answer:
The ratio of the length of \( PQ \) to the length of the latus rectum of the parabola is \( \frac{9}{8} \).