Find the locus of the middle points of chords of the parabola which
subtend a constant angle a at the vertex

Answer

To find the locus of the midpoints of chords of the parabola \( y^2 = 4Ax \) that subtend a constant angle \( \alpha \) at the vertex, we can follow these steps:

### Step 1: Identify Points on the Parabola
Let the points \( A \) and \( B \) on the parabola be represented in parametric form:
- Point \( A \) corresponds to parameter \( t_1 \): \( A(t_1) = (At_1^2, 2At_1) \)
- Point \( B \) corresponds to parameter \( t_2 \): \( B(t_2) = (At_2^2, 2At_2) \)

### Step 2: Find the Midpoint of the Chord
The midpoint \( M \) of the chord \( AB \) is given by:
\[ M = \left( \frac{At_1^2 + At_2^2}{2}, \frac{2At_1 + 2At_2}{2} \right) = \left( \frac{A(t_1^2 + t_2^2)}{2}, A(t_1 + t_2) \right) \]
Let \( H = \frac{A(t_1^2 + t_2^2)}{2} \) and \( K = A(t_1 + t_2) \).

### Step 3: Find the Slopes of Lines OA and OB
The slopes of the lines from the origin \( O(0,0) \) to points \( A \) and \( B \) are:
- Slope of \( OA \):
\[ m_1 = \frac{2At_1 - 0}{At_1^2 - 0} = \frac{2}{t_1} \]
- Slope of \( OB \):
\[ m_2 = \frac{2At_2 - 0}{At_2^2 - 0} = \frac{2}{t_2} \]

### Step 4: Use the Angle Condition
The angle \( \alpha \) between the two lines can be expressed using the tangent of the angle:
\[ \tan \alpha = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Substituting the slopes:
\[ \tan \alpha = \frac{\frac{2}{t_1} - \frac{2}{t_2}}{1 + \frac{4}{t_1 t_2}} = \frac{2(t_2 - t_1)}{t_1 t_2 + 4} \]

### Step 5: Rearranging the Equation
From the above equation, we can rearrange it to find a relationship between \( t_1 \) and \( t_2 \):
\[ t_2 - t_1 = \tan \alpha \left( \frac{t_1 t_2 + 4}{2} \right) \]

### Step 6: Use the Identity for \( t_1 + t_2 \) and \( t_1 t_2 \)
We know:
\[ (t_1 + t_2)^2 = t_1^2 + t_2^2 + 2t_1 t_2 \]
Substituting \( t_1 + t_2 = \frac{K}{A} \) and \( t_1^2 + t_2^2 = \frac{2H}{A} + 2t_1 t_2 \):
\[ \left( \frac{K}{A} \right)^2 = \frac{2H}{A} + 2t_1 t_2 \]

### Step 7: Solve for \( t_1 t_2 \)
From the above, we can express \( t_1 t_2 \) in terms of \( H \) and \( K \):
\[ t_1 t_2 = \frac{K^2}{2A^2} - \frac{H}{A} \]

### Step 8: Substitute Back into the Angle Condition
Now substitute \( t_1 t_2 \) back into the equation for \( t_2 - t_1 \) and simplify:
\[ (t_2 - t_1)^2 = \tan^2 \alpha \left( 4 + t_1 t_2 \right) \]
Substituting \( t_1 t_2 \):
\[ (t_2 - t_1)^2 = \tan^2 \alpha \left( 4 + \left( \frac{K^2}{2A^2} - \frac{H}{A} \right) \right) \]

### Step 9: Final Locus Equation
After simplifying, we arrive at the locus of the midpoint \( M(H, K) \):
\[ \tan^2 \alpha (8A^2 + K^2 - 2AH)^2 = 16A^2(4AH - K^2) \]

This is the required locus of the midpoints of the chords of the parabola that subtend a constant angle \( \alpha \) at the vertex.