Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola:
(i) `y^(2)=12x` (ii) `y^(2)=10x` (iii)`3y^(2)=8x`

Explanation

To solve the problem of finding the coordinates of the focus and vertex, the equations of the directrix and the axis, and the length of the latus rectum for the given parabolas, we will follow these steps for each parabola.

### (i) For the parabola \( y^2 = 12x \)

1. **Identify the standard form**: The equation \( y^2 = 12x \) is in the form \( y^2 = 4ax \).
- Here, \( 4a = 12 \) implies \( a = 3 \).

2. **Find the vertex**: The vertex of the parabola \( y^2 = 4ax \) is at the origin.
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