The equations of two waves given as `x = a cos (omega t = delta)` and `y = a cos (omega t + alpha)`, where `delta = alpha + pi/2`, then resultant wave represent:

Options

A. a circle (c.w)
B. A circle (a.c.w)
C. an Ellipse (c.w)
D. an ellipse (a.c.w)

Answer

To solve the problem, we need to analyze the equations of the two waves given as:

1. \( x = A \cos(\omega t + \delta) \)
2. \( y = A \cos(\omega t + \alpha) \)

where \( \delta = \alpha + \frac{\pi}{2} \).

### Step-by-step Solution:

**Step 1: Substitute the value of \( \delta \)**

From the problem, we know that:

\[ \delta = \alpha + \frac{\pi}{2} \]

Substituting this into the equation for \( x \):

\[ x = A \cos(\omega t + \alpha + \frac{\pi}{2}) \]

**Step 2: Simplify the equation for \( x \)**

Using the trigonometric identity \( \cos(\theta + \frac{\pi}{2}) = -\sin(\theta) \), we can rewrite \( x \):

\[ x = A \cos(\omega t + \alpha + \frac{\pi}{2}) = -A \sin(\omega t + \alpha) \]

**Step 3: Write the equations for both waves**

Now we have:

1. \( x = -A \sin(\omega t + \alpha) \)
2. \( y = A \cos(\omega t + \alpha) \)

**Step 4: Square both equations and add them**

To find the resultant wave, we square both equations:

\[ x^2 = A^2 \sin^2(\omega t + \alpha) \]
\[ y^2 = A^2 \cos^2(\omega t + \alpha) \]

Adding these two equations:

\[ x^2 + y^2 = A^2 \sin^2(\omega t + \alpha) + A^2 \cos^2(\omega t + \alpha) \]

**Step 5: Use the Pythagorean identity**

Using the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):

\[ x^2 + y^2 = A^2(1) = A^2 \]

**Step 6: Identify the shape of the resultant wave**

The equation \( x^2 + y^2 = A^2 \) represents a circle of radius \( A \).

**Step 7: Determine the direction of the wave**

To determine the direction (clockwise or anticlockwise), we analyze the parametric equations:

- For \( x = -A \sin(\omega t + \alpha) \)
- For \( y = A \cos(\omega t + \alpha) \)

As \( \omega t \) increases, \( \sin(\omega t + \alpha) \) increases from 0 to 1, causing \( x \) to go from 0 to -A, while \( y \) goes from A to 0. This indicates that the motion is in the anticlockwise direction.

### Final Answer:

The resultant wave represents a circle in the anticlockwise direction.

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