`x - t` equation of a particle executing SHM is
`x = Acos (omega t - 45^(@))`
Find the point from where particle starts its journey and the direction of its initial velocity.

Explanation

To solve the problem step by step, we will analyze the given equation of motion for the particle executing Simple Harmonic Motion (SHM) and determine the starting point and direction of the initial velocity.

### Step 1: Understand the given equation
The equation of motion for the particle is given as:
\[ x = A \cos(\omega t - 45^\circ) \]
Here, \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( t \) is the time.

### Step 2: Find the position at \( t = 0 \)
To find the starting point of the particle, we need to evaluate the position \( x \) when \( t = 0 \):
\[ x(0) = A \cos(\omega \cdot 0 - 45^\circ) \]
This simplifies to:
\[ x(0) = A \cos(-45^\circ) \]
Using the property of cosine, we know that \( \cos(-\theta) = \cos(\theta) \):
\[ x(0) = A \cos(45^\circ) \]
Since \( \cos(45^\circ) = \frac{1}{\sqrt{2}} \):
\[ x(0) = A \cdot \frac{1}{\sqrt{2}} = \frac{A}{\sqrt{2}} \]

### Step 3: Determine the direction of initial velocity
Next, we need to find the initial velocity of the particle. The velocity \( v \) in SHM is given by the derivative of the position with respect to time:
\[ v = \frac{dx}{dt} = \frac{d}{dt}(A \cos(\omega t - 45^\circ)) \]
Using the chain rule, we differentiate:
\[ v = -A \omega \sin(\omega t - 45^\circ) \]
Now, we evaluate the velocity at \( t = 0 \):
\[ v(0) = -A \omega \sin(\omega \cdot 0 - 45^\circ) \]
This simplifies to:
\[ v(0) = -A \omega \sin(-45^\circ) \]
Using the property of sine, we know that \( \sin(-\theta) = -\sin(\theta) \):
\[ v(0) = -A \omega (-\sin(45^\circ)) = A \omega \sin(45^\circ) \]
Since \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \):
\[ v(0) = A \omega \cdot \frac{1}{\sqrt{2}} = \frac{A \omega}{\sqrt{2}} \]

### Step 4: Determine the direction of the initial velocity
The initial velocity \( v(0) = \frac{A \omega}{\sqrt{2}} \) is positive, indicating that the particle is moving in the positive direction along the x-axis.

### Summary of Results
- The particle starts its journey at the position \( x = \frac{A}{\sqrt{2}} \).
- The direction of the initial velocity is positive.