If PQ is a focal chord of the parabola `y^(2)=4ax` with focus at s, then `(2SP.SQ)/(SP+SQ)=`

Options

A. a
B. 2a
C. 4a
D. `a^(2)`

Answer

To solve the problem, we need to find the value of \(\frac{2SP \cdot SQ}{SP + SQ}\) for the focal chord \(PQ\) of the parabola \(y^2 = 4ax\) with focus at \(S\).

### Step-by-step Solution:

1. **Identify the Focus of the Parabola**:
The given parabola is \(y^2 = 4ax\). The focus of this parabola is at the point \(S(a, 0)\).

**Hint**: Recall that for a parabola in the form \(y^2 = 4ax\), the focus is located at \((a, 0)\).

2. **Understanding Focal Chords**:
A focal chord is a line segment that passes through the focus of the parabola. Let \(P\) and \(Q\) be the endpoints of the focal chord.

3. **Using the Property of Focal Chords**:
For any focal chord \(PQ\) of the parabola \(y^2 = 4ax\), the following property holds:
\[ \frac{1}{SP} + \frac{1}{SQ} = \frac{2}{l} \]
where \(l\) is the length of the semi-latus rectum. The length of the semi-latus rectum for the parabola \(y^2 = 4ax\) is \(2a\).

**Hint**: Remember that the semi-latus rectum is a fixed distance related to the parabola's focus.

4. **Substituting the Length of the Semi-latus Rectum**:
Since the semi-latus rectum \(l = 2a\), we can substitute this into the equation:
\[ \frac{1}{SP} + \frac{1}{SQ} = \frac{2}{2a} = \frac{1}{a} \]

5. **Finding the Values of \(SP\) and \(SQ\)**:
Let \(SP = p\) and \(SQ = q\). Then we have:
\[ \frac{1}{p} + \frac{1}{q} = \frac{1}{a} \]
This can be rewritten as:
\[ \frac{p + q}{pq} = \frac{1}{a} \]
Cross-multiplying gives:
\[ a(p + q) = pq \]

6. **Rearranging the Equation**:
Rearranging the equation gives:
\[ pq = ap + aq \]

7. **Finding the Value of \(\frac{2pq}{p + q}\)**:
Now, we need to find:
\[ \frac{2pq}{p + q} \]
From our previous step, we know \(pq = a(p + q)\). Substituting this into our expression gives:
\[ \frac{2pq}{p + q} = \frac{2a(p + q)}{p + q} = 2a \]

### Final Answer:
Thus, the value of \(\frac{2SP \cdot SQ}{SP + SQ}\) is \(2a\).