Let O be the origin, and let A (1, 0), B (0, 1) be two points. If P (x, y) is a point such that `xygt0` and `x+ylt1` then:

Options

A. P lies either inside `DeltaOAB` or in third quadrant
B. P cannot be inside `DeltaOAB`
C. P lies inside the `DeltaOAB`
D. None of these

Answer

To solve the problem, we need to analyze the conditions given for the point P(x, y) in relation to the points A(1, 0) and B(0, 1).

### Step-by-Step Solution:

1. **Identify the Points and Conditions:**
- We have the origin O(0, 0), point A(1, 0), and point B(0, 1).
- The point P(x, y) must satisfy the conditions: \( x > 0 \), \( y > 0 \) (which means P is in the first quadrant) and \( x + y < 1 \).

2. **Understanding the Line x + y = 1:**
- The line defined by the equation \( x + y = 1 \) intersects the x-axis at A(1, 0) and the y-axis at B(0, 1).
- The area below this line and above the axes forms a triangle with vertices O(0, 0), A(1, 0), and B(0, 1).

3. **Determine the Region for P(x, y):**
- Since \( x + y < 1 \), point P must lie below the line \( x + y = 1 \).
- Given that \( x > 0 \) and \( y > 0 \), point P must also lie in the first quadrant.

4. **Conclusion:**
- Therefore, the point P(x, y) lies inside the triangle formed by the points O(0, 0), A(1, 0), and B(0, 1).
- Since the conditions also allow for \( x < 0 \) and \( y < 0 \) (which would place P in the third quadrant), we can conclude that P can also lie in the third quadrant if both x and y are negative.

### Final Answer:
The point P(x, y) lies inside the triangle OAB in the first quadrant and can also lie in the third quadrant if both coordinates are negative.