The curve represents by the equation`x^(2)/(sinsqrt2-cossqrt3)+y^(2)/(sinsqrt3-cossqrt2)=1` is

Options

A. (a) an ellipse with foci on X-axis
B. (b) an ellipse on focii Y-axis
C. (c) a hyperbola with foci on X-axis
D. (d) an hyperbola with foci on Y-axis

Answer

To determine the nature of the curve represented by the equation

\[ \frac{x^2}{\sin(\sqrt{2}) - \cos(\sqrt{3})} + \frac{y^2}{\sin(\sqrt{3}) - \cos(\sqrt{2})} = 1, \]

we will analyze the denominators and check if they satisfy the conditions for an ellipse.

### Step 1: Calculate \(\sin(\sqrt{2}) - \cos(\sqrt{3})\)

1. **Find \(\sin(\sqrt{2})\)**:
- Using a calculator, we find \(\sqrt{2} \approx 1.414\).
- Therefore, \(\sin(\sqrt{2}) \approx \sin(1.414) \approx 0.9877\).

2. **Find \(\cos(\sqrt{3})\)**:
- Using a calculator, we find \(\sqrt{3} \approx 1.732\).
- Therefore, \(\cos(\sqrt{3}) \approx \cos(1.732) \approx -0.1874\).

3. **Calculate \(\sin(\sqrt{2}) - \cos(\sqrt{3})\)**:
\[ \sin(\sqrt{2}) - \cos(\sqrt{3}) \approx 0.9877 - (-0.1874) = 0.9877 + 0.1874 = 1.1751. \]

### Step 2: Calculate \(\sin(\sqrt{3}) - \cos(\sqrt{2})\)

1. **Find \(\sin(\sqrt{3})\)**:
- \(\sin(\sqrt{3}) \approx \sin(1.732) \approx 0.9878\).

2. **Find \(\cos(\sqrt{2})\)**:
- \(\cos(\sqrt{2}) \approx \cos(1.414) \approx 0.1559\).

3. **Calculate \(\sin(\sqrt{3}) - \cos(\sqrt{2})\)**:
\[ \sin(\sqrt{3}) - \cos(\sqrt{2}) \approx 0.9878 - 0.1559 = 0.8319. \]

### Step 3: Analyze the equation

Now we can rewrite the equation with the calculated values:

\[ \frac{x^2}{1.1751} + \frac{y^2}{0.8319} = 1. \]

### Step 4: Determine the nature of the curve

The equation is in the standard form of an ellipse:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \]

where \(a^2 = 1.1751\) and \(b^2 = 0.8319\). Since \(a^2 > b^2\), the major axis is along the x-axis.

### Conclusion

Thus, the curve represented by the given equation is an **ellipse** with its major axis along the x-axis.