Let us define a region R is xy-plane as a set of points (x,y) satisfying `[x^2]=[y]` (where [x] denotes greatest integer `le x)`,then the region R defines

Options

A. a parabola whose axis is horizontal
B. a parabola whose axis is vertical
C. integer point of the parabola `y=x^2`
D. None of the above

Answer

To solve the problem, we need to analyze the given equation \( [x^2] = [y] \), where \([x]\) denotes the greatest integer less than or equal to \(x\).

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation states that the greatest integer less than or equal to \(x^2\) is equal to the greatest integer less than or equal to \(y\). This means that for any integer \(n\), we have:
\[ n \leq x^2 < n + 1 \quad \text{and} \quad n \leq y < n + 1 \]
This implies that both \(x^2\) and \(y\) lie within the same integer range.

2. **Finding the Range for \(y\)**:
Let's analyze the ranges for different values of \(n\):
- For \(n = 0\):
\[ 0 \leq x^2 < 1 \implies 0 \leq x < 1 \quad \text{and} \quad 0 \leq y < 1 \]
This gives us the rectangle defined by \((0,0)\) to \((1,1)\).

- For \(n = 1\):
\[ 1 \leq x^2 < 2 \implies 1 \leq |x| < \sqrt{2} \quad \text{and} \quad 1 \leq y < 2 \]
This gives us two intervals for \(x\): \([- \sqrt{2}, -1)\) and \((1, \sqrt{2})\) with \(y\) ranging from 1 to 2.

- For \(n = 2\):
\[ 2 \leq x^2 < 3 \implies \sqrt{2} \leq |x| < \sqrt{3} \quad \text{and} \quad 2 \leq y < 3 \]
This gives us two intervals for \(x\): \([- \sqrt{3}, -\sqrt{2})\) and \((\sqrt{2}, \sqrt{3})\) with \(y\) ranging from 2 to 3.

3. **Generalizing the Pattern**:
Continuing this process, we can see that for each integer \(n\):
- The values of \(y\) will always be in the range \([n, n+1)\).
- The corresponding values of \(x\) will be in the intervals \([- \sqrt{n+1}, -\sqrt{n})\) and \([\sqrt{n}, \sqrt{n+1})\).

4. **Visualizing the Region**:
The region \(R\) defined by these inequalities forms a series of horizontal strips in the \(xy\)-plane, where each strip corresponds to an integer \(n\) and contains segments of parabolas defined by \(y = x^2\).

5. **Conclusion**:
The region \(R\) does not correspond to a single parabola but rather consists of multiple segments of parabolas for each integer \(n\). Therefore, the correct option is:
- **Option 4: None of the above**.