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: Equation of the Chord
The equation of a chord of the hyperbola \( xy = c^2 \) can be expressed in the slope-intercept form. Given that the slope of the chord is \( m \), we can write the equation of the chord as:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is a point on the hyperbola.

### Step 2: Points on the Hyperbola
From the hyperbola equation \( xy = c^2 \), we can express \( y_1 \) in terms of \( x_1 \):
\[ y_1 = \frac{c^2}{x_1} \]
Substituting this into the chord equation gives:
\[ y - \frac{c^2}{x_1} = m(x - x_1) \]

### Step 3: Finding the Midpoint
Let the endpoints of the chord be \( (x_1, y_1) \) and \( (x_2, y_2) \). The midpoint \( (H, K) \) of the chord can be expressed as:
\[ H = \frac{x_1 + x_2}{2}, \quad K = \frac{y_1 + y_2}{2} \]
Using the hyperbola condition for both points, we have:
\[ y_1 = \frac{c^2}{x_1}, \quad y_2 = \frac{c^2}{x_2} \]
Thus,
\[ K = \frac{c^2}{2} \left( \frac{1}{x_1} + \frac{1}{x_2} \right) \]

### Step 4: Relationship Between \( x_1 \) and \( x_2 \)
Since the chord is parallel and has a fixed slope \( m \), we can express \( x_2 \) in terms of \( x_1 \):
\[ y_2 - y_1 = m(x_2 - x_1) \]
Substituting \( y_1 \) and \( y_2 \):
\[ \frac{c^2}{x_2} - \frac{c^2}{x_1} = m(x_2 - x_1) \]
This can be rearranged to find a relationship between \( x_1 \) and \( x_2 \).

### Step 5: Locus of Midpoints
After manipulating the equations, we can derive the locus of the midpoints. The relationship will yield a linear equation in terms of \( H \) and \( K \).

### Final Result
The locus of the midpoints of the parallel chords with slope \( m \) of the hyperbola \( xy = c^2 \) is given by:
\[ K = \frac{c^2}{m} H \]
This represents a straight line.