The locus of the mid - points of the parallel chords with slope m of the rectangular hyperbola `xy=c^(2)` is

Options

A. `y+mx=0`
B. `y-mx=0`
C. `my-x=0`
D. `my+x=0`

Answer

To find the locus of the midpoints of the parallel chords with slope \( m \) of the rectangular hyperbola \( xy = c^2 \), we can follow these steps:

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

### Step 2: Write the equation of the chord
The equation of the chord with midpoint \( P(h, k) \) can be expressed using the midpoint formula:
\[ kx + hy = 2hk \]

### Step 3: Incorporate the slope of the chord
Given that the slope of the chord is \( m \), we can relate the coordinates \( h \) and \( k \) using the slope:
\[ \frac{-k}{h} = m \quad \Rightarrow \quad -k = mh \quad \Rightarrow \quad k + mh = 0 \]

### Step 4: Rearranging the equation
From the equation \( k + mh = 0 \), we can express \( k \) in terms of \( h \):
\[ k = -mh \]

### Step 5: Substitute \( k \) into the chord equation
Now substitute \( k = -mh \) into the chord equation:
\[ (-mh)x + hy = 2h(-mh) \]
This simplifies to:
\[ -mhx + hy = -2mh^2 \]

### Step 6: Rearranging to find the locus
Rearranging the above equation gives:
\[ hy - mhx + 2mh^2 = 0 \]

### Step 7: Factor out \( h \)
Factoring out \( h \) from the equation:
\[ h(y - mx + 2mh) = 0 \]

### Step 8: Identify the locus
For the locus, we consider the case when \( h \neq 0 \):
\[ y - mx + 2mh = 0 \]
This represents a family of lines with slope \( m \).

### Step 9: Final form of the locus
Thus, the locus of the midpoints of the parallel chords is given by:
\[ y + mx = 0 \]

### Conclusion
The locus of the midpoints of the parallel chords with slope \( m \) of the rectangular hyperbola \( xy = c^2 \) is:
\[ y + mx = 0 \]