Final answer:
Kepler's first law states that planetary orbits are elliptical with the Sun at one focus. To find the perihelion and aphelion distances, use the equations: Perihelion distance = semimajor axis - eccentricity * semimajor axis and Aphelion distance = semimajor axis + eccentricity * semimajor axis. The equation that represents the orbital path of the asteroid is x^2 / a^2 + y^2 / b^2 = 1.
Step-by-step explanation:
Kepler's first law states that the planetary orbits are elliptical with the Sun at one focus. The perihelion distance is the distance between the Sun and the closest point on the asteroid's orbit, and the aphelion distance is the distance between the Sun and the farthest point. The semiminor axis is half the length of the major axis, which is equal to half the difference between the aphelion and perihelion distances.
To find the perihelion distance, we subtract the eccentricity multiplied by the semimajor axis from the semimajor axis:
Perihelion distance = semimajor axis - eccentricity * semimajor axis.
To find the aphelion distance, we add the eccentricity multiplied by the semimajor axis to the semimajor axis:
Aphelion distance = semimajor axis + eccentricity * semimajor axis.
The equation that represents the orbital path of the asteroid is given by:
x^2 / a^2 + y^2 / b^2 = 1, where a is the semimajor axis and b is the semiminor axis.