If one end of a focal chord of the parabola `y^(2)=4ax` be `(at^(2),2at)`, then the coordinates of its other end is

Options

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

Answer

To find the coordinates of the other end of the focal chord of the parabola given one end as \((at^2, 2at)\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given point**:
The coordinates of one end of the focal chord are given as \((at^2, 2at)\).

2. **Understand the properties of focal chords**:
In a parabola of the form \(y^2 = 4ax\), if one end of a focal chord corresponds to the parameter \(t\), the other end corresponds to the parameter \(t_1\) such that:
\[ t_1 = -\frac{1}{t} \]

3. **Find the coordinates of the other end**:
Using the parameter \(t_1\), we can find the coordinates of the other end of the focal chord:
- The x-coordinate is given by:
\[ x_1 = at_1^2 = a\left(-\frac{1}{t}\right)^2 = \frac{a}{t^2} \]
- The y-coordinate is given by:
\[ y_1 = 2at_1 = 2a\left(-\frac{1}{t}\right) = -\frac{2a}{t} \]

4. **Combine the coordinates**:
Therefore, the coordinates of the other end of the focal chord are:
\[ \left(\frac{a}{t^2}, -\frac{2a}{t}\right) \]

### Final Answer:
The coordinates of the other end of the focal chord are \(\left(\frac{a}{t^2}, -\frac{2a}{t}\right)\).