A particle is acted simultaneously by mutually perpendicular simple harmonic motions `x=acos 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 of finding the trajectory of a particle acted upon by two mutually perpendicular simple harmonic motions given by the equations \( x = a \cos(\omega t) \) and \( y = a \sin(\omega t) \), we can follow these steps:

### Step 1: Write down the equations
We have the two equations:
1. \( x = a \cos(\omega t) \) (Equation 1)
2. \( y = a \sin(\omega t) \) (Equation 2)

### Step 2: Square both equations
Next, we square both equations:
- From Equation 1:
\[ x^2 = (a \cos(\omega t))^2 = a^2 \cos^2(\omega t) \]
- From Equation 2:
\[ y^2 = (a \sin(\omega t))^2 = a^2 \sin^2(\omega t) \]

### Step 3: Add the squared equations
Now, we add the two squared equations:
\[ x^2 + y^2 = a^2 \cos^2(\omega t) + a^2 \sin^2(\omega t) \]

### Step 4: Factor out the common term
We can factor out \( a^2 \):
\[ x^2 + y^2 = a^2 (\cos^2(\omega t) + \sin^2(\omega t)) \]

### Step 5: Use the Pythagorean identity
Using the Pythagorean identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \), we simplify the equation:
\[ x^2 + y^2 = a^2 \cdot 1 \]
Thus, we have:
\[ x^2 + y^2 = a^2 \]

### Step 6: Identify the trajectory
The equation \( x^2 + y^2 = a^2 \) represents a circle with radius \( a \).

### Conclusion
Therefore, the trajectory of the particle is a circle with radius \( a \).