If the line segment joining P(2, 3) and Q(5, 7) subtends a right angle at R(x, y) and the area of `DeltaPQR=2` sq. units, then the maximum number of such points R is xy - plane are

Answer

To solve the problem, we need to find the maximum number of points \( R(x, y) \) in the xy-plane such that the line segment joining points \( P(2, 3) \) and \( Q(5, 7) \) subtends a right angle at \( R \) and the area of triangle \( PQR \) is 2 square units.

### Step-by-Step Solution:

1. **Find the coordinates of points P and Q**:
- \( P(2, 3) \)
- \( Q(5, 7) \)

2. **Calculate the distance \( PQ \)**:
\[ PQ = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. **Use the area formula for triangle \( PQR \)**:
The area of triangle \( PQR \) can be expressed as:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base \( PQ = 5 \) and the area is given as 2 square units. Let the height from point \( R \) to line \( PQ \) be \( h \).
\[ 2 = \frac{1}{2} \times 5 \times h \]
Simplifying this gives:
\[ h = \frac{2 \times 2}{5} = \frac{4}{5} \]

4. **Determine the maximum height**:
The maximum possible height \( H \) from point \( R \) to line \( PQ \) can be calculated using the formula for the altitude from a point to a line. The maximum height occurs when \( R \) is directly perpendicular to line \( PQ \).
- The maximum height \( H \) is given by:
\[ H = \frac{PQ}{2} = \frac{5}{2} \]

5. **Find the coordinates of point R**:
Since we know that the area condition gives us a height of \( \frac{4}{5} \), we can find the points \( R \) that satisfy both the right angle condition and the area condition.

6. **Equation of line PQ**:
The slope of line \( PQ \) is:
\[ m = \frac{7 - 3}{5 - 2} = \frac{4}{3} \]
The equation of line \( PQ \) in point-slope form (using point \( P \)):
\[ y - 3 = \frac{4}{3}(x - 2) \]
Rearranging gives:
\[ 4x - 3y + 1 = 0 \]

7. **Find the line perpendicular to PQ through R**:
The slope of the line perpendicular to \( PQ \) is \( -\frac{3}{4} \). The equation of the line through point \( R(x, y) \) can be expressed as:
\[ y - y_1 = -\frac{3}{4}(x - x_1) \]

8. **Determine the intersection points**:
The intersection of this line with the line \( PQ \) will give us the coordinates of points \( R \) that satisfy the right angle condition.

9. **Count the possible points**:
Since the height \( h = \frac{4}{5} \) is less than the maximum height \( H = \frac{5}{2} \), there will be two points above and two points below line \( PQ \) that satisfy both conditions. Thus, the maximum number of such points \( R \) is:
\[ \text{Maximum number of points } R = 4 \]

### Final Answer:
The maximum number of such points \( R \) in the xy-plane is **4**.