x^{2} –3y + 2x + 3 = 0

x^{2} + 2x + 1 = 3y – 2 ⇒ (x + 1)^{2} = 3(y -2/3).

Its axis is the line x + 1 = 0.

It's vertex is (-1, 2/3).

It's focus is x + 1 = 0, y - 2/3 = 3/4 i.e., (-1, 17/12).

It's directrix is the line y - 2/3 = - 3/4, i.e., y + 1/12 = 0.

It's latus rectum is the line y - 2/3 = 3/4. i.e., y - 17/12 = 0.

Extremities of it,s latus rectum are x + 1 = 3/2, y - 2/3, i.e., (1/2, 17/12).

Length of it's latus rectum is 3 units.