Find the coordinates of the focus and the vertex, the equations of the

Solution:

Given equation :

$3 x^{2}=8 y$

$x^{2}=\frac{8}{3} y$

Comparing the given equation with parabola having an equation,

$x^{2}=4 a y$

$\cdot 4 a=\frac{8}{3}$

- $a=\frac{2}{3}$

Focus: $F(0, a)=F\left(0, \frac{2}{3}\right)$

Vertex : $A(0,0)=A(0,0)$

Equation of the directrix: $y+a=0$

$y+\frac{2}{3}=0$

$y=-\frac{2}{3}$

Lenth of latusrectum :

$4 a=\frac{8}{3}$