Locus of the mid-points of all chords of the parabola `y^(2)=4ax` which are drwan through the vertex is a parabola, then its latus rectum is

Options

A. `(a)/(2)`
B. a
C. 2a
D. `(3a)/(2)`

Explanation

To solve the problem, we need to find the locus of the midpoints of all chords of the parabola \( y^2 = 4ax \) that are drawn through the vertex. We will derive the equation of this locus and then determine the latus rectum of the resulting parabola.

### Step-by-Step Solution:

1. **Understand the Parabola**:
The given parabola is \( y^2 = 4ax \). The vertex of this parabola is at the origin (0, 0).

2. **Chords through the Vertex**:
Any chord of the parabola that passes through the vertex can be represented by two points on the parabola. Let's denote these points as \( P(t_1) \) and \( P(t_2) \), where:
\[ P(t_1) = (at_1^2, 2at_1) \quad \text{and} \quad P(t_2) = (at_2^2, 2at_2) \]

3. **Midpoint of the Chord**:
The midpoint \( M \) of the chord \( P(t_1)P(t_2) \) is given by:
\[ M = \left( \frac{at_1^2 + at_2^2}{2}, \frac{2at_1 + 2at_2}{2} \right) = \left( \frac{a(t_1^2 + t_2^2)}{2}, a(t_1 + t_2) \right) \]

4. **Expressing \( t_1 + t_2 \) and \( t_1^2 + t_2^2 \)**:
Let \( s = t_1 + t_2 \) and \( p = t_1 t_2 \). We can express \( t_1^2 + t_2^2 \) as:
\[ t_1^2 + t_2^2 = (t_1 + t_2)^2 - 2t_1t_2 = s^2 - 2p \]

5. **Substituting into the Midpoint Coordinates**:
The coordinates of the midpoint \( M \) can now be rewritten as:
\[ M = \left( \frac{a(s^2 - 2p)}{2}, as \right) \]

6. **Finding the Locus**:
To find the locus, we eliminate \( s \) and \( p \). From the midpoint coordinates, we have:
\[ x = \frac{a(s^2 - 2p)}{2} \quad \text{and} \quad y = as \]
Rearranging gives:
\[ s = \frac{y}{a} \]
Substituting this into the equation for \( x \):
\[ x = \frac{a\left(\left(\frac{y}{a}\right)^2 - 2p\right)}{2} \]
Simplifying gives:
\[ x = \frac{y^2}{2a} - ap \]
To express \( p \) in terms of \( x \) and \( y \), we can rearrange:
\[ ap = \frac{y^2}{2a} - x \quad \Rightarrow \quad p = \frac{y^2}{2a^2} - \frac{x}{a} \]

7. **Final Equation of the Locus**:
Since \( p \) is a product of roots, we can express the locus as:
\[ y^2 = 2ax \]
This is the equation of a parabola.

8. **Finding the Latus Rectum**:
The standard form of a parabola is \( y^2 = 4px \). Here, we have \( y^2 = 2ax \), which can be rewritten as:
\[ y^2 = 4\left(\frac{a}{2}\right)x \]
Thus, the latus rectum \( L \) is given by \( 4p = 4 \times \frac{a}{2} = 2a \).

### Conclusion:
The latus rectum of the parabola formed by the locus of the midpoints of the chords is \( 2a \).