Two paricles A and B start simultaneously from the same point and move in a horizontal plane. A has an initial velocity `u_(1)` due east and acceleration `a_(1)` due north. B has an intial velocity `u_(2)` due north and acceleration `a_(2)` due east.

Options

A. Their paths must intersect at some point
B. They must collide at some point
C. They will collide only if `a_1u_1=a_2u_2`
D. If `u_1 gt u_2` and `a_1 lt a_2`, the particles will have the same speed at some point of time

Explanation

To solve the problem of two particles A and B moving in a horizontal plane with given initial velocities and accelerations, we can break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Motion of Particle A:**
- Particle A starts with an initial velocity \( u_1 \) due east and has an acceleration \( a_1 \) due north.
- The displacement of particle A after time \( t \) can be expressed using the equation of motion:
\[ \vec{S_A} = u_1 \hat{i} t + \frac{1}{2} a_1 \hat{j} t^2 \]
- Here, \( \hat{i} \) represents the eastward direction and \( \hat{j} \) represents the northward direction.

2. **Understanding the Motion of Particle B:**
- Particle B starts with an initial velocity \( u_2 \) due north and has an acceleration \( a_2 \) due east.
- The displacement of particle B after time \( t \) can be expressed as:
\[ \vec{S_B} = u_2 \hat{j} t + \frac{1}{2} a_2 \hat{i} t^2 \]

3. **Setting Up the Displacement Equations:**
- For particle A:
\[ \vec{S_A} = u_1 t \hat{i} + \frac{1}{2} a_1 t^2 \hat{j} \]
- For particle B:
\[ \vec{S_B} = \frac{1}{2} a_2 t^2 \hat{i} + u_2 t \hat{j} \]

4. **Finding Conditions for Intersection:**
- For the paths of particles A and B to intersect, their displacements must be equal at some time \( t \):
\[ u_1 t = \frac{1}{2} a_2 t^2 \quad \text{(1)} \]
\[ \frac{1}{2} a_1 t^2 = u_2 t \quad \text{(2)} \]

5. **Dividing the Equations:**
- From equations (1) and (2), we can divide them to eliminate \( t \):
\[ \frac{u_1 t}{\frac{1}{2} a_2 t^2} = \frac{\frac{1}{2} a_1 t^2}{u_2 t} \]
- Simplifying gives:
\[ \frac{u_1}{\frac{1}{2} a_2 t} = \frac{\frac{1}{2} a_1 t}{u_2} \]
- This leads to:
\[ \frac{u_1}{u_2} = \frac{a_2}{a_1} \]
- Rearranging gives:
\[ u_1 a_1 = u_2 a_2 \quad \text{(3)} \]

6. **Analyzing the Options:**
- **Option 1:** Their paths must intersect at some point - **Correct** (as derived).
- **Option 2:** They must collide at some point - **Incorrect** (they will collide only if \( u_1 a_1 = u_2 a_2 \)).
- **Option 3:** If \( u_1 > u_2 \) and \( a_1 < a_2 \), they will have the same speed at some point - **Correct** (from the relationship derived).

### Final Conclusion:
The correct options are that their paths must intersect at some point, and if \( u_1 > u_2 \) and \( a_1 < a_2 \), they will have the same speed at some point.