If b and k are segments of a focal chord of the parabola `y^(2)= 4ax`, then k =

Options

A. `(ab)/(b-a)`
B. `(b)/(b-a)`
C. `(a)/(b-a)`
D. `(ab)/(a-b)`

Explanation

To solve the problem, we need to find the value of \( k \) given that \( b \) and \( k \) are segments of a focal chord of the parabola defined by the equation \( y^2 = 4ax \).

### Step-by-Step Solution:

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

2. **Parametric Representation**:
The points on the parabola can be represented parametrically as:
\[ P(t) = (at^2, 2at) \]
where \( t \) is a parameter.

3. **Endpoints of the Focal Chord**:
For a focal chord, the endpoints can be represented as:
\[ P(t_1) = (at_1^2, 2at_1) \quad \text{and} \quad P(t_2) = (at_2^2, 2at_2) \]
where \( t_1 \cdot t_2 = -1 \) (a property of focal chords).

4. **Finding the Length of the Segments**:
The distance \( SP \) from the focus \( S(a, 0) \) to the point \( P(t) \) is calculated using the distance formula:
\[ SP = \sqrt{(at^2 - a)^2 + (2at - 0)^2} \]
Simplifying this gives:
\[ SP = \sqrt{a^2(t^2 - 1)^2 + (2at)^2} = \sqrt{a^2((t^2 - 1)^2 + 4t^2)} \]

5. **Setting Up the Equation**:
Let \( SP = b \). Thus, we have:
\[ b = a \sqrt{(t^2 - 1)^2 + 4t^2} \]

6. **Finding the Length of the Other Segment**:
Similarly, we can find the distance \( SQ \) from the focus \( S(a, 0) \) to the other endpoint of the focal chord:
\[ SQ = \sqrt{(at_2^2 - a)^2 + (2at_2 - 0)^2} \]
Since \( t_2 = -\frac{1}{t_1} \), we can express \( SQ \) in terms of \( t_1 \).

7. **Using the Focal Chord Property**:
From the property of focal chords, we know:
\[ t_1 t_2 = -1 \implies t_2 = -\frac{1}{t_1} \]

8. **Equating the Segments**:
We can express \( k \) in terms of \( b \) using the relationship derived from the distances:
\[ k = \frac{b a}{b - a} \]

### Final Answer:
Thus, the value of \( k \) is:
\[ k = \frac{ba}{b - a} \]