A particle is acted simultaneously by mutually perpendicular simple harmonic motions x=a cos`omegat` and `y=asinomegat`. The trajectory of motion of the particle will be

Options

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

Explanation

To solve the problem, we need to analyze the given equations of motion for the particle:

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

2. **Squaring Both Equations**:
- Square both equations to eliminate the trigonometric functions:
\[ x^2 = a^2 \cos^2(\omega t) \]
\[ y^2 = a^2 \sin^2(\omega t) \]

3. **Adding the Two Equations**:
- Now, add the two equations:
\[ x^2 + y^2 = a^2 \cos^2(\omega t) + a^2 \sin^2(\omega t) \]

4. **Using Trigonometric Identity**:
- We know from trigonometric identities that:
\[ \cos^2(\theta) + \sin^2(\theta) = 1 \]
- Therefore, we can substitute this identity into our equation:
\[ x^2 + y^2 = a^2 (\cos^2(\omega t) + \sin^2(\omega t)) = a^2 \cdot 1 = a^2 \]

5. **Final Equation**:
- The resulting equation is:
\[ x^2 + y^2 = a^2 \]
- This is the standard form of the equation of a circle with radius \( a \).

6. **Conclusion**:
- Thus, the trajectory of the motion of the particle is a circle.

**Final Answer**: The trajectory of motion of the particle will be a circle.

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