If b,c are the segments of a focal chord of the parabola `y^2 = 4ax`, then c is equal to

Options

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

Answer

To solve the problem, we need to find the value of \( c \) given that \( b \) and \( c \) 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 equation of the parabola is \( y^2 = 4ax \). This is a standard form of a parabola that opens to the right.

2. **Identify the Length of the Latus Rectum**:
The length of the latus rectum (L) of the parabola \( y^2 = 4ax \) is given by \( L = 4a \). The semi-latus rectum (half of the length) is therefore \( 2a \).

3. **Relationship Between Segments of Focal Chord**:
For a focal chord, the semi-latus rectum is the harmonic mean of the segments \( b \) and \( c \). The relationship can be expressed as:
\[ 2a = \frac{2bc}{b+c} \]

4. **Rearranging the Equation**:
To find \( c \), we can rearrange the equation:
\[ 2a(b+c) = 2bc \]
Simplifying gives:
\[ 2ab + 2ac = 2bc \]

5. **Isolating \( c \)**:
Rearranging the equation to isolate \( c \):
\[ 2ac = 2bc - 2ab \]
Dividing through by \( 2a \) (assuming \( a \neq 0 \)):
\[ c = \frac{bc - ab}{a} \]

6. **Final Expression for \( c \)**:
We can express \( c \) in terms of \( a \) and \( b \):
\[ c = \frac{ab}{b - a} \]

### Conclusion:
Thus, the value of \( c \) is given by:
\[ c = \frac{ab}{b - a} \]