A particle is subjected simultaneously to two SHMs, one along the x - axis and the other along the y - axis. The two vibrations are in phase and have unequal amplitudes. The particle will execute

Options

A. Straight line motion
B. Circular motion
C. Elliptic motion
D. Parabolic motion

Answer

To solve the problem, we need to analyze the motion of a particle subjected to two simultaneous simple harmonic motions (SHMs) along the x-axis and y-axis. The key points to consider are:

1. **Understanding SHM**: Simple harmonic motion can be described mathematically. For the x-axis, we can express the SHM as:
\[ x(t) = A_1 \sin(\omega t) \]
where \( A_1 \) is the amplitude of the motion along the x-axis and \( \omega \) is the angular frequency.

2. **SHM along the y-axis**: Similarly, for the y-axis, the SHM can be expressed as:
\[ y(t) = A_2 \sin(\omega t) \]
where \( A_2 \) is the amplitude of the motion along the y-axis.

3. **Relationship between x and y**: Since both SHMs are in phase, we can express the relationship between x and y. By eliminating time \( t \), we can relate x and y directly:
\[ \frac{y}{A_2} = \sin(\omega t) \quad \text{and} \quad \frac{x}{A_1} = \sin(\omega t) \]
This implies:
\[ \frac{y}{A_2} = \frac{x}{A_1} \]
Rearranging gives:
\[ y = \frac{A_2}{A_1} x \]

4. **Identifying the type of motion**: The equation \( y = \frac{A_2}{A_1} x \) represents a straight line through the origin (0,0) with a slope of \( \frac{A_2}{A_1} \). This indicates that the particle will move along a straight line in the xy-plane.

5. **Conclusion**: Since the particle's motion can be described by a linear relationship between x and y, the particle will execute straight line motion.

### Final Answer:
The particle will execute **straight line motion**.