A right-opening parabola has latus rectum length 6 cm, while an upward-opening parabola has latus rectum length 10 cm. Both parabolas have their vertices at the origin and are drawn to the same scale.
(i) Determine the value of the parameter a for each of the two parabolas.
(ii) Write the standard equation of each parabola.
(iii) Using a suitable numerical comparison, state which of the two parabolas appears wider at the vertex.