Prove that the locus of the middle points of all chords of the parabola `y^2 = 4ax` passing through the vertex is the parabola `y^2 = 2ax`.

Options

A. `y^(2)=8x`
B. `y^(2)=2x`
C. `x^(2)+4y^(2)=16`
D. `x^(2)=2y`

Answer

To prove that the locus of the midpoints of all chords of the parabola \( y^2 = 4ax \) passing through the vertex is the parabola \( y^2 = 2ax \), we can follow these steps:

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

### Step 2: Parametric Form of the Parabola
The points on the parabola can be expressed in parametric form:
- Let \( P(t_1) = (at_1^2, 2at_1) \) and \( Q(t_2) = (at_2^2, 2at_2) \) be two points on the parabola.

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

### Step 4: Express \( t_1 + t_2 \) and \( t_1^2 + t_2^2 \)
Using the identity \( t_1^2 + t_2^2 = (t_1 + t_2)^2 - 2t_1t_2 \), let \( t_1 + t_2 = k \) and \( t_1 t_2 = m \). Then:
\[ t_1^2 + t_2^2 = k^2 - 2m \]

### Step 5: Substitute into Midpoint Coordinates
Substituting \( t_1^2 + t_2^2 \) into the midpoint coordinates:
\[ M\left( \frac{a(k^2 - 2m)}{2}, ak \right) \]

### Step 6: Eliminate \( m \)
Since the chord passes through the vertex (0,0), we can use the fact that the product \( t_1 t_2 \) can be expressed in terms of the slope of the chord. For a chord through the vertex, we can set \( m = 0 \):
\[ M\left( \frac{ak^2}{2}, ak \right) \]

### Step 7: Relate \( k \) to \( y \)
Let \( y = ak \) then \( k = \frac{y}{a} \). Substitute this into the x-coordinate:
\[ x = \frac{a\left(\frac{y}{a}\right)^2}{2} = \frac{y^2}{2a} \]

### Step 8: Rearranging the Equation
Rearranging gives:
\[ y^2 = 2ax \]

### Conclusion
Thus, we have shown that the locus of the midpoints of all chords of the parabola \( y^2 = 4ax \) passing through the vertex is indeed the parabola \( y^2 = 2ax \).