Normals drawn to the hyperbola `xy=2` at the point `P(t_1)` meets the hyperbola again at `Q(t_2)`, then minimum distance between the point P and Q is

Answer

To find the minimum distance between the points \( P(t_1) \) and \( Q(t_2) \) on the hyperbola defined by the equation \( xy = 2 \), we will follow these steps:

### Step 1: Identify the coordinates of points P and Q
The coordinates of point \( P(t_1) \) on the hyperbola can be expressed as:
\[ P(t_1) = \left( \sqrt{2} t_1, \frac{\sqrt{2}}{t_1} \right) \]
Similarly, the coordinates of point \( Q(t_2) \) are:
\[ Q(t_2) = \left( \sqrt{2} t_2, \frac{\sqrt{2}}{t_2} \right) \]

### Step 2: Establish the relationship between \( t_1 \) and \( t_2 \)
The relationship between \( t_1 \) and \( t_2 \) when the normal at \( P(t_1) \) meets the hyperbola again at \( Q(t_2) \) is given by:
\[ t_2 = -\frac{1}{t_1^3} \]

### Step 3: Calculate the distance \( PQ \)
The distance \( PQ \) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of \( P \) and \( Q \):
\[ d = \sqrt{\left( \sqrt{2} t_2 - \sqrt{2} t_1 \right)^2 + \left( \frac{\sqrt{2}}{t_2} - \frac{\sqrt{2}}{t_1} \right)^2} \]
This simplifies to:
\[ d = \sqrt{2} \sqrt{(t_2 - t_1)^2 + \left( \frac{1}{t_2} - \frac{1}{t_1} \right)^2} \]

### Step 4: Substitute \( t_2 \) into the distance formula
Substituting \( t_2 = -\frac{1}{t_1^3} \):
\[ d = \sqrt{2} \sqrt{\left(-\frac{1}{t_1^3} - t_1\right)^2 + \left(-t_1^3 - \frac{1}{t_1}\right)^2} \]

### Step 5: Simplify the expression
Now we will simplify the expression inside the square root. Let’s denote:
\[ A = -\frac{1}{t_1^3} - t_1 \]
\[ B = -t_1^3 - \frac{1}{t_1} \]
Then:
\[ d = \sqrt{2} \sqrt{A^2 + B^2} \]

### Step 6: Find the minimum value of the distance
To minimize \( d \), we can use the Cauchy-Schwarz inequality:
\[ \frac{t_1^2 + \frac{1}{t_1^2}}{2} \geq 1 \]
This implies:
\[ t_1^2 + \frac{1}{t_1^2} \geq 2 \]
Thus, the minimum value of \( d \) occurs when \( t_1^2 + \frac{1}{t_1^2} = 2 \), which gives:
\[ d_{\text{min}} = \sqrt{2} \cdot \sqrt{2} = 2 \]

### Final Result
The minimum distance between points \( P \) and \( Q \) is:
\[ \boxed{4} \]