An ellipse is a geometric shape that resembles an elongated circle. The equation \(\left(\frac{x}{b}\right)^2 + \left(\frac{y}{c}\right)^2 = 1\) as derived in the exercise is a standard representation of an ellipse.
- Centered at the origin.
- With a semi-major axis of length \(b\) in the x-direction
- And a semi-minor axis of length \(c\) in the y-direction
The significant property of an ellipse is that for any point on it, the sum of the distances from two fixed points (foci) is constant.
This property helps us understand the particle's path in the exercise. By eliminating the parameter \( t \) from the position function, we can visualize the particle's motion more broadly as a distinct, recognizable shape.
The particle's path is clearly illustrated as it travels, demonstrating how mathematics and geometry intersect to provide a clear picture of motion.