Find the locus of the middle points of chords of the parabola which
are of given length l

Answer

To find the locus of the midpoints of chords of a parabola with a given length \( l \), we can follow these steps:

### Step 1: Define the Parabola and Points on it
We start with the standard equation of the parabola:
\[ y^2 = 4ax \]
Let the points on the parabola be \( A(t_1) \) and \( B(t_2) \) where:
\[ A(t_1) = (at_1^2, 2at_1) \quad \text{and} \quad B(t_2) = (at_2^2, 2at_2) \]

### Step 2: Find the Midpoint of the Chord
The midpoint \( M(h, k) \) of the chord \( AB \) can be calculated as:
\[ h = \frac{at_1^2 + at_2^2}{2} = \frac{a(t_1^2 + t_2^2)}{2} \]
\[ k = \frac{2at_1 + 2at_2}{2} = a(t_1 + t_2) \]

### Step 3: Calculate the Length of the Chord
The length \( l \) of the chord \( AB \) can be calculated using the distance formula:
\[ l = \sqrt{(at_2^2 - at_1^2)^2 + (2at_2 - 2at_1)^2} \]
This simplifies to:
\[ l = \sqrt{a^2(t_2^2 - t_1^2)^2 + 4a^2(t_2 - t_1)^2} \]
Factoring out \( a^2 \):
\[ l = a \sqrt{(t_2^2 - t_1^2)^2 + 4(t_2 - t_1)^2} \]

### Step 4: Simplify the Expression
Using the identity \( (t_2^2 - t_1^2) = (t_2 - t_1)(t_2 + t_1) \):
\[ l = a \sqrt{(t_2 - t_1)^2(t_2 + t_1)^2 + 4(t_2 - t_1)^2} \]
Let \( d = t_2 - t_1 \) and \( s = t_2 + t_1 \):
\[ l = a |d| \sqrt{s^2 + 4} \]

### Step 5: Solve for \( d \)
From the equation above, we can express \( d \):
\[ d = \frac{l}{a \sqrt{s^2 + 4}} \]

### Step 6: Substitute Back to Find the Locus
Substituting \( s = 2h/a \) and \( d = t_2 - t_1 \):
\[ t_2 + t_1 = \frac{2h}{a} \]
Using the relation \( t_2 - t_1 \) and substituting into the equation:
\[ (t_2 - t_1)^2 = \left(\frac{l}{a \sqrt{\left(\frac{2h}{a}\right)^2 + 4}}\right)^2 \]
This leads to a relationship between \( h \) and \( k \).

### Step 7: Final Equation of the Locus
After simplifying, we arrive at the equation of the locus:
\[ x + \frac{l^2}{4a^2} = 0 \]
This represents the locus of the midpoints of the chords of the parabola.