A particel which is simultaneously subjected to two perpendicular simple harmonic motions represented by `x=a_(1)cosomegat` and `y=a_(2)cos2omegat` traces a curve given by :

Options

A.
B.
C.
D.

Explanation

To solve the problem of finding the curve traced by a particle subjected to two perpendicular simple harmonic motions represented by \( x = a_1 \cos(\omega t) \) and \( y = a_2 \cos(2\omega t) \), we will follow these steps:

### Step 1: Understand the given equations
We have two equations:
- \( x = a_1 \cos(\omega t) \)
- \( y = a_2 \cos(2\omega t) \)

### Step 2: Express \( \cos(2\omega t) \) in terms of \( \cos(\omega t) \)
Using the double angle formula for cosine:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
we can write:
\[ \cos(2\omega t) = 2\cos^2(\omega t) - 1 \]

### Step 3: Substitute \( \cos(\omega t) \) in terms of \( x \)
From the first equation, we can express \( \cos(\omega t) \):
\[ \cos(\omega t) = \frac{x}{a_1} \]
Now substituting this into the expression for \( \cos(2\omega t) \):
\[ \cos(2\omega t) = 2\left(\frac{x}{a_1}\right)^2 - 1 = \frac{2x^2}{a_1^2} - 1 \]

### Step 4: Substitute into the equation for \( y \)
Now we substitute this back into the equation for \( y \):
\[ y = a_2 \left(2\left(\frac{x}{a_1}\right)^2 - 1\right) \]
This simplifies to:
\[ y = a_2 \left(2\frac{x^2}{a_1^2} - 1\right) = \frac{2a_2}{a_1^2} x^2 - a_2 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ y + a_2 = \frac{2a_2}{a_1^2} x^2 \]
This can be rewritten as:
\[ \frac{2a_2}{a_1^2} x^2 = y + a_2 \]

### Step 6: Identify the type of curve
The equation \( \frac{2a_2}{a_1^2} x^2 = y + a_2 \) is in the form of a parabola, specifically:
\[ x^2 = \frac{a_1^2}{2a_2}(y + a_2) \]
This shows that the curve is a parabola that opens upwards.

### Conclusion
The curve traced by the particle is a parabola symmetric about the y-axis.

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