Question

Hey can someone please help explain and answer this 1 question. There's a picture. Thank you!

Answer 1

We'll first clear a few points.
1. A hyperbola with horizontal axis and centred on origin (i.e. foci are centred on the x-axis) has equation
x^2/a^2-y^2/b^2=1
(check: when y=0, x=+/- a, the vertices)
The corresponding hyperbola with vertical axis centred on origin has equation
y^2/a^2-x^2/b^2=1
(check: when x=0, y=+/- a, the vertices).

The co-vertex is the distance b in the above formula, such that 
the distance of the foci from the origin, c satisfies c^2=a^2+b^2.
The rectangle with width a and height b has diagonals which are the asymptotes of the hyperbola.

We're given vertex = +/- 3, and covertex=+/- 5.
And since vertices are situated at (3,0), and (-3,0), they are along the x-axis.
So the equation must start with
x^2/3^2.

It will be good practice for you to sketch all four hyperbolas given in the choices to fully understand the basics of a hyperbola.