Robotics & Automated Manufacturing (CNC Laser Alignment)

A high-precision 5-axis robotic laser cutter must trim a straight edge along an angled automotive chassis segment. The software commands the laser focal head to move in a straight line, starting from an approach point at $C = (20, 10, 45)$ (in millimeters relative to the machine base origin) and targeting a destination coordinate at $D = (80, -20, 15)$.

1. Find the parametric equations of the laser toolpath.
2. Convert the toolpath equations into symmetric form.
3. If a physical safety clamp is fixed exactly at the coordinate where $x = 50$, determine the $y$ and $z$ coordinates of the laser beam as it crosses this plane to evaluate if it will hit the fixture.