The locus of midpoints of chords of the parabola `y^(2)=4ax` which are parallel to line y =mx +c is

Options

A. `y=(2a)/(m)`
B. `x=(2a)/(m)`
C. y=2a
D. x=2a

Answer

To find the locus of midpoints of chords of the parabola \( y^2 = 4ax \) that are parallel to the line \( y = mx + c \), we can follow these steps:

### Step 1: Understand the parabola and the chord
The given parabola is \( y^2 = 4ax \). The general form of a chord of the parabola can be expressed using the midpoint formula. Let the midpoint of the chord be \( (h, k) \).

### Step 2: Equation of the chord
The equation of the chord of the parabola that passes through the point \( (h, k) \) can be derived from the chord of contact formula. The chord of contact for the point \( (h, k) \) is given by:
\[ ky = 2a(x + h) \]
This can be rearranged to:
\[ ky - 2ax - 2ah = 0 \]

### Step 3: Slope of the chord
Since the chord is parallel to the line \( y = mx + c \), it will have the same slope \( m \). The slope of the line represented by the equation \( ky - 2ax - 2ah = 0 \) can be obtained by rewriting it in slope-intercept form:
\[ ky = 2ax + 2ah \implies y = \frac{2a}{k}x + \frac{2ah}{k} \]
From this, we see that the slope is \( \frac{2a}{k} \).

### Step 4: Set slopes equal
Since the slopes are equal, we can set:
\[ \frac{2a}{k} = m \]
This leads to:
\[ k = \frac{2a}{m} \]

### Step 5: Locus of midpoints
Now, we have expressed \( k \) in terms of \( h \) and \( m \). The coordinates of the midpoint are \( (h, k) \), where \( k = \frac{2a}{m} \). The locus of the midpoints is independent of \( h \) and can be expressed as:
\[ y = \frac{2a}{m} \]

### Conclusion
Thus, the locus of the midpoints of the chords of the parabola \( y^2 = 4ax \) that are parallel to the line \( y = mx + c \) is given by:
\[ y = \frac{2a}{m} \]