P is a variable point on `xy = 1` in the first quadrant and R is a variable point on the positive x- axis . If rhombus OPQR ( in order ) is completed (where O is the origin ) , then find the locus of point Q.

Answer

To find the locus of point Q in the rhombus OPQR where P is a variable point on the hyperbola \(xy = 1\) in the first quadrant and R is a variable point on the positive x-axis, we can follow these steps:

### Step 1: Define the Points
Let:
- \( O = (0, 0) \) (the origin)
- \( P = (a, \frac{1}{a}) \) where \( a > 0 \) (since P lies on the hyperbola \(xy = 1\))
- \( R = (b, 0) \) where \( b > 0 \) (since R lies on the positive x-axis)

### Step 2: Understand the Properties of the Rhombus
In a rhombus, the diagonals bisect each other at right angles. Thus, the midpoint of diagonal PR should be the same as the midpoint of diagonal OQ.

### Step 3: Find the Coordinates of Q
Let the coordinates of Q be \( Q = (h, k) \). The midpoint of PR is:
\[ M_{PR} = \left( \frac{a + b}{2}, \frac{\frac{1}{a} + 0}{2} \right) = \left( \frac{a + b}{2}, \frac{1}{2a} \right) \]
The midpoint of OQ is:
\[ M_{OQ} = \left( \frac{0 + h}{2}, \frac{0 + k}{2} \right) = \left( \frac{h}{2}, \frac{k}{2} \right) \]
Setting these midpoints equal gives us:
\[ \frac{h}{2} = \frac{a + b}{2} \quad \text{and} \quad \frac{k}{2} = \frac{1}{2a} \]

### Step 4: Solve for h and k
From the first equation:
\[ h = a + b \]
From the second equation:
\[ k = \frac{1}{a} \]

### Step 5: Substitute for a
Since \( P \) lies on the hyperbola \( xy = 1 \), we have:
\[ k = \frac{1}{a} \implies a = \frac{1}{k} \]
Substituting \( a \) into the equation for \( h \):
\[ h = \frac{1}{k} + b \]

### Step 6: Express b in terms of h and k
From \( h = \frac{1}{k} + b \), we can express \( b \) as:
\[ b = h - \frac{1}{k} \]

### Step 7: Find the Locus of Q
Since \( R \) is on the x-axis, we have \( b > 0 \). Thus:
\[ h - \frac{1}{k} > 0 \implies h > \frac{1}{k} \]
This can be rearranged to give:
\[ hk > 1 \]
Thus, the locus of point Q is given by:
\[ hk = 1 \]

### Conclusion
The locus of point Q is the hyperbola defined by the equation:
\[ xy = 1 \]
where \( x \) and \( y \) are the coordinates of point Q.