If `t` is the parameter for one end of a focal chord of the parabola `y^2 =4ax,` then its length is :

Options

A. `a(t+1/t)^(2)`
B. `a(t-1/t)^(2)`
C. `a(t+1/t)`
D. `a(t-1/t)`

Answer

To find the length of a focal chord of the parabola \( y^2 = 4ax \) when one end is defined by the parameter \( t \), we can follow these steps:

### Step 1: Identify the points on the parabola
The coordinates of the point on the parabola corresponding to the parameter \( t \) are given by:
\[ P(t) = (at^2, 2at) \]
Let the other end of the focal chord be defined by the parameter \( t_1 \). The coordinates of this point are:
\[ Q(t_1) = (at_1^2, 2at_1) \]

### Step 2: Use the property of focal chords
For a parabola, the product of the parameters of the endpoints of a focal chord is always \(-1\):
\[ t \cdot t_1 = -1 \quad \Rightarrow \quad t_1 = -\frac{1}{t} \]

### Step 3: Calculate the coordinates of point Q
Substituting \( t_1 \) into the coordinates of point \( Q \):
\[ Q\left(-\frac{1}{t}\right) = \left(a\left(-\frac{1}{t}\right)^2, 2a\left(-\frac{1}{t}\right)\right) = \left(\frac{a}{t^2}, -\frac{2a}{t}\right) \]

### Step 4: Find the length of the focal chord PQ
The length \( PQ \) can be calculated using the distance formula:
\[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points \( P \) and \( Q \):
\[ PQ = \sqrt{\left(\frac{a}{t^2} - at^2\right)^2 + \left(-\frac{2a}{t} - 2at\right)^2} \]

### Step 5: Simplify the expression
1. **For the x-coordinates**:
\[ x_2 - x_1 = \frac{a}{t^2} - at^2 = a\left(\frac{1}{t^2} - t^2\right) = a\left(\frac{1 - t^4}{t^2}\right) \]
Therefore,
\[ (x_2 - x_1)^2 = a^2\left(\frac{1 - t^4}{t^2}\right)^2 = \frac{a^2(1 - t^4)^2}{t^4} \]

2. **For the y-coordinates**:
\[ y_2 - y_1 = -\frac{2a}{t} - 2at = -2a\left(\frac{1}{t} + t\right) = -2a\left(\frac{1 + t^2}{t}\right) \]
Therefore,
\[ (y_2 - y_1)^2 = 4a^2\left(\frac{1 + t^2}{t}\right)^2 = \frac{4a^2(1 + t^2)^2}{t^2} \]

### Step 6: Combine the results
Putting it all together:
\[ PQ = \sqrt{\frac{a^2(1 - t^4)^2}{t^4} + \frac{4a^2(1 + t^2)^2}{t^2}} \]
Factoring out \( a^2 \):
\[ PQ = a \sqrt{\frac{(1 - t^4)^2}{t^4} + \frac{4(1 + t^2)^2}{t^2}} \]

### Step 7: Simplify further
Combining the fractions:
\[ PQ = a \sqrt{\frac{(1 - t^4)^2 + 4t^2(1 + t^2)^2}{t^4}} \]
This gives us the length of the focal chord in terms of \( a \) and \( t \).

### Final Result
The length of the focal chord is:
\[ PQ = a \sqrt{(t^2 + 1)^2} \]
Thus, the length of the focal chord is:
\[ PQ = a(t^2 + 1) \]