Find the locus of the middle points of chords of the parabola which
pass through the focus.

Answer

To find the locus of the midpoints of chords of the parabola that pass through the focus, we will follow these steps:

### Step 1: Understand the Parabola and its Focus
The standard equation of the parabola is given by:
\[ y^2 = 4ax \]
The focus of this parabola is at the point \( (a, 0) \).

### Step 2: Parametric Representation of Points on the Parabola
Let the points on the parabola be represented parametrically. For two points \( A \) and \( B \) on the parabola, we can use the parameters \( t_1 \) and \( t_2 \):
- Coordinates of point \( A \):
\[ A(t_1) = (at_1^2, 2at_1) \]
- Coordinates of point \( B \):
\[ B(t_2) = (at_2^2, 2at_2) \]

### Step 3: Equation of the Chord AB
The equation of the chord \( AB \) can be derived using the two-point form of the line. The equation is:
\[ y(t_1 + t_2) = 2x + 2a t_1 t_2 \]

### Step 4: Midpoint of the Chord
Let the midpoint of the chord \( AB \) be \( M(h, k) \). The coordinates of the midpoint can be expressed as:
- \( h = \frac{at_1^2 + at_2^2}{2} \)
- \( k = \frac{2at_1 + 2at_2}{2} = a(t_1 + t_2) \)

### Step 5: Condition for the Chord to Pass through the Focus
Since the chord passes through the focus \( (a, 0) \), we substitute \( (x, y) = (a, 0) \) into the chord equation:
\[ 0 = 2a + 2a t_1 t_2 \]
This simplifies to:
\[ 1 + t_1 t_2 = 0 \]
Thus, we have:
\[ t_1 t_2 = -1 \]

### Step 6: Find \( t_1 + t_2 \)
Using the identity:
\[ (t_1 + t_2)^2 = t_1^2 + t_2^2 + 2t_1 t_2 \]
We can express \( t_1^2 + t_2^2 \) in terms of \( h \):
Since \( t_1 t_2 = -1 \), we can write:
\[ t_1^2 + t_2^2 = (t_1 + t_2)^2 - 2(-1) = (t_1 + t_2)^2 + 2 \]

### Step 7: Substitute into the Midpoint Coordinates
From the midpoint coordinates:
- \( h = \frac{a(t_1^2 + t_2^2)}{2} \)
Substituting \( t_1^2 + t_2^2 \):
\[ h = \frac{a((t_1 + t_2)^2 + 2)}{2} \]

### Step 8: Express \( k \) in terms of \( h \)
Since \( k = a(t_1 + t_2) \), we can express \( t_1 + t_2 \) as:
\[ t_1 + t_2 = \frac{k}{a} \]
Substituting this into the equation for \( h \):
\[ h = \frac{a\left(\left(\frac{k}{a}\right)^2 + 2\right)}{2} = \frac{k^2}{2a} + a \]

### Step 9: Rearranging to Find the Locus
Rearranging gives us:
\[ k^2 = 2ah - 2a^2 \]
This can be rewritten in standard form:
\[ k^2 = 2a(h - a) \]

### Conclusion: Locus of Midpoints
Thus, the locus of the midpoints of the chords of the parabola that pass through the focus is given by:
\[ y^2 = 2a(x - a) \]