The coordinates of a particle moving in a plane are given by $x(t) = a \cos (pt)$ and $y(t) = b \sin (pt)$ where $a, b (< a)$ and $p$ are positive constants of appropriate dimensions. Then

Step 1: Identify the path of the particle.

Eliminate the parameter $t$ from the given parametric equations.

$$x(t) = a \cos(pt)$$ $$y(t) = b \sin(pt)$$ $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2(pt) + \sin^2(pt)$$ $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

This represents an ellipse, so statement (1) is correct.

Step 2: Determine the velocity and acceleration vectors.

Differentiate the position vector $\vec{r}(t) = (x(t), y(t))$ with respect to time.

$$\vec{v}(t) = \frac{d\vec{r}}{dt} = (-ap \sin(pt), bp \cos(pt))$$ $$\vec{a}(t) = \frac{d\vec{v}}{dt} = (-ap^2 \cos(pt), -bp^2 \sin(pt))$$

Step 3: Evaluate velocity and acceleration at $t = \frac{\pi}{2p}$.

Substitute $t = \frac{\pi}{2p}$ into the vector expressions.

$$\vec{v}\left(\frac{\pi}{2p}\right) = (-ap, 0)$$ $$\vec{a}\left(\frac{\pi}{2p}\right) = (0, -bp^2)$$

Calculate the dot product of the two vectors.

$$\vec{v} \cdot \vec{a} = 0$$

Since the dot product is zero, the velocity and acceleration are normal to each other, so statement (2) is correct.

Step 4: Analyze the direction of the acceleration vector.

Express the acceleration vector in terms of the position vector.

$$\vec{a}(t) = -p^2 (a \cos(pt), b \sin(pt))$$ $$\vec{a}(t) = -p^2 \vec{r}(t)$$

The acceleration is directed towards the origin (the center of the ellipse) rather than a focus, so statement (3) is incorrect.

Step 5: Calculate the distance traveled from $t = 0$ to $t = \frac{\pi}{2p}$.

Set up the arc length integral for the path.

$$L = \int_{0}^{\frac{\pi}{2p}} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$ $$L = \int_{0}^{\frac{\pi}{2p}} \sqrt{a^2 p^2 \sin^2(pt) + b^2 p^2 \cos^2(pt)} \, dt$$

Substitute $u = pt$ to simplify the integral.

$$L = \int_{0}^{\pi/2} \sqrt{a^2 \sin^2 u + b^2 \cos^2 u} \, du$$

Since $b < a$, this integral is not equal to $a$, so statement (4) is incorrect.

Answer: The correct statements are:

1. The path of the particle is an ellipse.
2. At $t = \frac{\pi}{2p}$, the velocity and acceleration are normal to each other.