A particle is subjected to two mutually perpendicualr simple harmonic motions such that its x and y-coordinates are given by
`x=sinomegat`, `y=2cosomegat`
The path of the particle will be :

Options

A. an ellipse
B. a straight line
C. a parabola
D. a circle

Answer

To determine the path of a particle subjected to two mutually perpendicular simple harmonic motions, we start with the given equations for the x and y coordinates:

1. **Given Equations**:
- \( x = \sin(\omega t) \)
- \( y = 2 \cos(\omega t) \)

2. **Expressing Cosine in Terms of y**:
We can express \( \cos(\omega t) \) in terms of \( y \):
\[ \cos(\omega t) = \frac{y}{2} \]
This gives us our first equation.

3. **Using the Pythagorean Identity**:
We know from trigonometry that:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Applying this identity, we can square both the expressions for \( x \) and \( y \):
\[ \sin^2(\omega t) + \cos^2(\omega t) = 1 \]

4. **Substituting the Values**:
Substitute \( \sin(\omega t) = x \) and \( \cos(\omega t) = \frac{y}{2} \) into the identity:
\[ x^2 + \left(\frac{y}{2}\right)^2 = 1 \]

5. **Simplifying the Equation**:
Now, simplify the equation:
\[ x^2 + \frac{y^2}{4} = 1 \]

6. **Rearranging to Standard Form**:
We can rearrange this equation to match the standard form of an ellipse:
\[ \frac{x^2}{1} + \frac{y^2}{4} = 1 \]

7. **Identifying the Shape**:
The equation \( \frac{x^2}{1} + \frac{y^2}{4} = 1 \) is the standard form of an ellipse, where \( a^2 = 1 \) and \( b^2 = 4 \).

8. **Conclusion**:
Therefore, the path of the particle is an ellipse.

### Final Answer:
The path of the particle will be an ellipse.