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This is a mathematical word problem written in German about two planes moving towards each other.

### Translation of the problem:
"A flight controller recognizes on the radar screen that two planes are approaching each other.

- Plane **GLX 1378** is at position \((112, 10)\), while plane **Jet 2222** is at position \((-119, 17)\).
- One minute later, **GLX 1378** is at position \((-113, 1)\), and **Jet 2222** is at \((-218, 6)\).

The planes are moving in straight lines. The coordinates are given in kilometers.

Judge whether the flight controller needs to intervene to prevent a potential collision."

### Problem Breakdown:
- We're given the initial and one-minute-later positions of two planes.
- The positions are in kilometers.
- We are tasked with determining if these two planes are on a collision course.

### Approach:
1. **Calculate the velocities** of both planes based on their change in position over one minute.
2. **Determine the equations of the lines** along which both planes are traveling.
3. **Check if the two lines intersect**, and if so, where and when. If they intersect within a short timeframe, the planes may collide.

Let me start by calculating the velocities of the planes.The velocities of the two planes are as follows:

- **GLX 1378**: \((-225, -9)\) km/min.
- **Jet 2222**: \((-99, -11)\) km/min.

This means:
- GLX 1378 is moving 225 km west and 9 km south per minute.
- Jet 2222 is moving 99 km west and 11 km south per minute.

Next, we can check if their paths will intersect by determining if they are on a collision course. This requires comparing their trajectories and seeing if they cross at the same point at the same time.

I'll calculate the parametric equations for the paths of both planes and solve for the potential intersection point.The solution for the time when the planes' paths would intersect is approximately:

- \( t_{\text{GLX}} = -2.04 \) minutes
- \( t_{\text{Jet}} = -2.31 \) minutes

Since both times are negative, this means the planes would have intersected at that point in the past. Therefore, **the planes are not on a collision course** in the future, and no intervention is needed to prevent a collision.

Would you like further details, or do you have any questions?

Here are 5 related questions to expand on this:
1. How can we calculate the distance between two planes at any given time?
2. What factors should be considered for a safe distance between aircraft in practice?
3. How do you solve parametric equations for general motion problems in physics?
4. Can we calculate when the planes will be closest to each other in the future?
5. How would this scenario change if we considered acceleration?

**Tip**: In collision problems, checking whether the times are physically feasible (i.e., positive) is essential to ensure a valid prediction for future events.

Auf dem Radarschirm erkennt ein Fluglotse, dass sich zwei Flugzeuge nähern. Flugzeug GLX 1378 hat die Position (112, 10), während Flugzeug Jet 2222 die Position (-119, 17) hat. Eine Minute später hat GLX 1378 die Position (-113, 1) und Jet 2222 bei (-218, 6). Die Flugzeuge bewegen sich geradlinig. Die Koordinaten sind in Kilometern angegeben. Beurteilen Sie, ob der Fluglotse eingreifen muss, um eine mögliche Kollision zu verhindern.

Determine if two planes, GLX 1378 and Jet 2222, are on a collision course based on their positions over time. This problem involves calculating velocities, deriving linear equations, and solving for potential intersections in a real-world context. Suitable for high school students studying coordinate geometry and vector analysis.

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