The parametric coordinates of any point on the parabola `y^(2) = 4ax` can be

Options

A. `(at^(2),2at)`
B. `(at^(2),-2at)`
C. `(asin^(2)t,2asint)`
D. `(asint,2acost)`

Answer

To find the parametric coordinates of any point on the parabola given by the equation \( y^2 = 4ax \), we can derive the parametric equations step by step.

### Step-by-Step Solution:

1. **Understanding the Parabola**:
The equation of the parabola is \( y^2 = 4ax \). This is a standard form of a parabola that opens to the right.

2. **Choosing Parametric Coordinates**:
A common choice for parametric coordinates on this parabola is:
\[ x = at^2 \]
where \( t \) is a parameter.

3. **Finding the Corresponding y-coordinate**:
To find \( y \), we substitute \( x = at^2 \) into the parabola's equation:
\[ y^2 = 4a(at^2) = 4a^2t^2 \]
Taking the square root gives us:
\[ y = \pm 2at \]
Thus, the parametric coordinates of any point on the parabola can be expressed as:
\[ (x, y) = (at^2, \pm 2at) \]

4. **Verifying Other Parametric Forms**:
We can also check other forms of parametric equations:
- For \( x = a \sin^2(t) \):
\[ y^2 = 4a(a \sin^2(t)) = 4a^2 \sin^2(t) \]
Thus, \( y = \pm 2a \sin(t) \).
- For \( x = 2at \):
\[ y^2 = 4a(2at) = 8a^2t \]
This does not satisfy the original parabola equation.

5. **Conclusion**:
The valid parametric coordinates for the parabola \( y^2 = 4ax \) are:
- \( (at^2, 2at) \)
- \( (at^2, -2at) \)
- \( (a \sin^2(t), 2a \sin(t)) \)
- \( (a \sin^2(t), -2a \sin(t)) \)

### Final Answer:
The parametric coordinates of any point on the parabola \( y^2 = 4ax \) can be given by:
- \( (at^2, 2at) \)
- \( (at^2, -2at) \)
- \( (a \sin^2(t), 2a \sin(t)) \)
- \( (a \sin^2(t), -2a \sin(t)) \)