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

Answer

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

### Step 1: Understand the parabola and its parameters
The parabola \( y^2 = 4ax \) has its focus at the point \( (a, 0) \) and the directrix is the line \( x = -a \). The parameter \( t \) corresponds to a point on the parabola given by the coordinates \( (at^2, 2at) \).

### Step 2: Find the coordinates of the endpoints of the focal chord
For a focal chord, if one end is at \( (at^2, 2at) \), the other end can be determined using the property of focal chords in parabolas. The product of the parameters of the endpoints of a focal chord is always \(-1\). Therefore, if one end has the parameter \( t_1 = t \), the other end will have the parameter \( t_2 = -\frac{1}{t} \).

### Step 3: Calculate the coordinates of the second endpoint
Using \( t_2 = -\frac{1}{t} \), the coordinates of the second endpoint are:
\[ \left( a\left(-\frac{1}{t}\right)^2, 2a\left(-\frac{1}{t}\right) \right) = \left( \frac{a}{t^2}, -\frac{2a}{t} \right) \]

### Step 4: Find the length of the focal chord
The length of the focal chord can be calculated using the distance formula between the two endpoints:
\[ \text{Length} = \sqrt{ \left( at^2 - \frac{a}{t^2} \right)^2 + \left( 2at - \left(-\frac{2a}{t}\right) \right)^2 } \]

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

Now substituting these into the distance formula:
\[ \text{Length} = \sqrt{ \left( a\left(\frac{t^4 - 1}{t^2}\right) \right)^2 + \left( 2a\left(t + \frac{1}{t}\right) \right)^2 } \]

### Step 6: Further simplification
Calculating the squares:
1. The first term:
\[ \left( a\left(\frac{t^4 - 1}{t^2}\right) \right)^2 = a^2\frac{(t^4 - 1)^2}{t^4} \]
2. The second term:
\[ \left( 2a\left(t + \frac{1}{t}\right) \right)^2 = 4a^2\left(t + \frac{1}{t}\right)^2 \]

Combining these gives:
\[ \text{Length} = a\left(t + \frac{1}{t}\right)^2 \]

### Final Result
Thus, the length of the focal chord is:
\[ \text{Length} = a\left(t + \frac{1}{t}\right)^2 \]