The latus rectum of a parabola whose focal chord PSQ is such that `SP = 3` and `SQ = 2` is given by

Options

A. `6//5`
B. `12//5`
C. `24//5`
D. none

Answer

To find the latus rectum of the parabola given the lengths of the segments of the focal chord PSQ, we can follow these steps:

### Step 1: Understand the relationship between the segments and the latus rectum
The segments of the focal chord are given as \( SP = 3 \) and \( SQ = 2 \). The latus rectum \( LR \) is related to these segments through the harmonic mean.

### Step 2: Set up the equation using the harmonic mean
The relationship can be expressed as:
\[ \frac{1}{SP} + \frac{1}{SQ} = \frac{2}{\frac{LR}{2}} \]
Substituting the values of \( SP \) and \( SQ \):
\[ \frac{1}{3} + \frac{1}{2} = \frac{2}{\frac{LR}{2}} \]

### Step 3: Simplify the left-hand side
To combine the fractions on the left side, we need a common denominator:
\[ \frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6} \]
So, we have:
\[ \frac{5}{6} = \frac{4}{LR} \]

### Step 4: Cross-multiply to solve for \( LR \)
Cross-multiplying gives:
\[ 5 \cdot LR = 6 \cdot 4 \]
\[ 5 \cdot LR = 24 \]
Now, divide both sides by 5:
\[ LR = \frac{24}{5} \]

### Step 5: Conclusion
Thus, the latus rectum \( LR \) of the parabola is \( \frac{24}{5} \).

### Final Answer
The latus rectum of the parabola is \( \frac{24}{5} \).

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