Answer:
[tex]y = - \frac{3}{20} {x}^{2} [/tex]
Step-by-step explanation:
The given parabola has focus at:
[tex](0, - \frac{5}{3} )[/tex]
and directrix at
[tex]y = \frac{5}{3} [/tex]
This is a vertical parabola that opens downwards.
The equation is of the form;
[tex] {x}^{2} = - 4py[/tex]
p is the distance from the vertex to the directrix.
Since the vertex is at the origin, we have
[tex]p = \frac{5}{3} [/tex]
We plug this value into the equation to get:
[tex] {x}^{2} = - 4( \frac{5}{3} )y[/tex]
[tex] {x}^{2} = - \frac{20}{3} y[/tex]
We solve for y to obtain:
[tex]y = - \frac{3}{20} {x}^{2} [/tex]
The 3rd option is correct.
