Find the locus of the midpoint of chords of the parabola `y^2=4a x` that pass through the point `(3a ,a)dot`

Explanation

To find the locus of the midpoint of chords of the parabola \( y^2 = 4ax \) that pass through the point \( (3a, a) \), we can follow these steps:

### Step 1: Define the midpoint of the chord
Let the midpoint of the chord be \( (h, k) \).

### Step 2: Write the equation of the chord
The equation of the chord of the parabola \( y^2 = 4ax \) with midpoint \( (h, k) \) is given by:
\[ ky - 2a(x + h) = k^2 - 4ah \]

### Step 3: Substitute the point through which the chord passes
Since the chord passes through the point \( (3a, a) \), we substitute \( x = 3a \) and \( y = a \) into the chord equation:
\[ ka - 2a(3a + h) = k^2 - 4ah \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ ka - 6a^2 - 2ah = k^2 - 4ah \]
Rearranging gives:
\[ k^2 - ka + 4ah - 6a^2 = 0 \]

### Step 5: Rearranging to find the locus
This is a quadratic equation in \( k \). For \( k \) to have real values, the discriminant must be non-negative. The discriminant \( D \) of the quadratic equation \( k^2 - ka + (4ah - 6a^2) = 0 \) is given by:
\[ D = (-a)^2 - 4 \cdot 1 \cdot (4ah - 6a^2) \]
\[ D = a^2 - 4(4ah - 6a^2) \]
\[ D = a^2 - 16ah + 24a^2 \]
\[ D = 25a^2 - 16ah \]

### Step 6: Set the discriminant to be non-negative
For the locus, we need \( D \geq 0 \):
\[ 25a^2 - 16ah \geq 0 \]
This can be rearranged to:
\[ 25a^2 \geq 16ah \]
Dividing both sides by \( a \) (assuming \( a > 0 \)):
\[ 25a \geq 16h \]
Thus, we can express \( h \):
\[ h \leq \frac{25a}{16} \]

### Step 7: Expressing the locus in terms of \( k \)
Now substituting \( h \) back into the equation for \( k \):
Using \( k^2 - ka + 4ah - 6a^2 = 0 \) and solving for \( k \) gives us the relation of \( k \) in terms of \( h \).

### Final Locus Equation
The final locus of the midpoint \( (h, k) \) can be expressed as:
\[ k^2 - ak + 4ah - 6a^2 = 0 \]