If PSQ is focal chord of the parabola `y^2 = 8x` such that `SP=6`, then the length SQ is

Options

A. 3
B. 4
C. 6
D. none

Answer

To solve the problem, we need to find the length of segment SQ in the focal chord PSQ of the parabola \( y^2 = 8x \) given that the length of segment SP is 6.

### Step-by-Step Solution:

1. **Identify the parameters of the parabola**:
The given parabola is \( y^2 = 8x \). We can compare this with the standard form \( y^2 = 4ax \) to find \( a \).
\[ 4a = 8 \implies a = 2 \]

**Hint**: Remember that the focal length \( a \) is derived from the standard form of the parabola.

2. **Determine the semi-latus rectum**:
The length of the latus rectum of the parabola is given by \( 4a \).
\[ \text{Length of latus rectum} = 4a = 4 \times 2 = 8 \]
The semi-latus rectum \( m \) is half of this:
\[ m = \frac{8}{2} = 4 \]

**Hint**: The semi-latus rectum is important in defining the relationship between segments in a focal chord.

3. **Use the relationship in harmonic progression**:
Since PSQ is a focal chord, the segments PS, SQ, and the semi-latus rectum \( m \) are in harmonic progression. The condition for three lengths \( a, b, c \) to be in harmonic progression is given by:
\[ \frac{1}{PS} + \frac{1}{SQ} = \frac{2}{m} \]
Here, \( PS = 6 \) and \( m = 4 \).

**Hint**: Recall the formula for harmonic progression and how to substitute the known values.

4. **Substituting the known values**:
Substitute \( PS = 6 \) and \( m = 4 \) into the harmonic progression condition:
\[ \frac{1}{6} + \frac{1}{SQ} = \frac{2}{4} = \frac{1}{2} \]

**Hint**: Be careful with fractions; ensure you find a common denominator when solving.

5. **Solve for \( \frac{1}{SQ} \)**:
Rearranging the equation gives:
\[ \frac{1}{SQ} = \frac{1}{2} - \frac{1}{6} \]
To find a common denominator (which is 6):
\[ \frac{1}{2} = \frac{3}{6} \implies \frac{1}{SQ} = \frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \]

**Hint**: Simplifying fractions can help in finding the final answer more easily.

6. **Finding \( SQ \)**:
Taking the reciprocal gives:
\[ SQ = 3 \]

**Hint**: The reciprocal of a fraction gives the length of the segment you are looking for.

### Final Answer:
The length of segment \( SQ \) is \( 3 \).