`y^2=4ax` is a parabola . A line segment joining focus of parabola to any moving point `(at^2,2at)` is made . Then locus of mid-point of the line segment is a parabola with the directrix

Options

A. `x=a/2`
B. `x=0`
C. `x=-a/2`
D. `x=a`

Explanation

To solve the problem, we need to find the locus of the midpoint of the line segment joining the focus of the parabola \( y^2 = 4ax \) to a moving point on the parabola given by \( P(at^2, 2at) \).

### Step-by-Step Solution:

1. **Identify the Focus of the Parabola**:
The standard form of the parabola \( y^2 = 4ax \) has its focus at the point \( (a, 0) \).

2. **Define the Moving Point**:
The moving point \( P \) on the parabola can be represented as \( P(at^2, 2at) \).

3. **Find the Midpoint**:
Let the midpoint \( Q \) of the line segment joining the focus \( S(a, 0) \) and the point \( P(at^2, 2at) \) be given by:
\[ Q\left( \frac{a + at^2}{2}, \frac{0 + 2at}{2} \right) = Q\left( \frac{a(1 + t^2)}{2}, at \right) \]

4. **Express Coordinates of Midpoint**:
Let the coordinates of the midpoint \( Q \) be \( (h, k) \):
\[ h = \frac{a(1 + t^2)}{2}, \quad k = at \]

5. **Eliminate the Parameter \( t \)**:
From the equation \( k = at \), we can express \( t \) as:
\[ t = \frac{k}{a} \]
Substituting this into the equation for \( h \):
\[ h = \frac{a\left(1 + \left(\frac{k}{a}\right)^2\right)}{2} = \frac{a + \frac{k^2}{a}}{2} = \frac{a^2 + k^2}{2a} \]

6. **Rearranging the Equation**:
Rearranging the equation gives:
\[ 2ah = a^2 + k^2 \]
This can be rewritten as:
\[ k^2 = 2ah - a^2 \]

7. **Identify the Locus**:
The equation \( k^2 = 2ah - a^2 \) represents a parabola with its vertex at \( (a, 0) \) and opens to the right.

8. **Find the Directrix**:
The directrix of a parabola in the form \( k^2 = 4px \) is given by \( x = -p \). Here, we can see that \( 2a \) corresponds to \( 4p \), thus \( p = \frac{a}{2} \). Therefore, the directrix is:
\[ x = -\frac{a}{2} \]

### Final Answer:
The locus of the midpoint of the line segment is a parabola with the directrix given by:
\[ x = -\frac{a}{2} \]