If the telescope is 9 m deep and 12 m wide, how far is the focus from the vertex?

If the telescope has a parabolic surface, then we can use the equation of a parabola to solve this.

If the parabola is facing to the right with the vertex at the origin, the we have the points of the parabola from the given:
(9,6)
(9,-6)
(9,0)

For a parabola with the center at origin and facing to the right, we have the standard form:
y^2 = 4ax
a is the distance from the vertex to the focus

Choosing any of the coordinates, say (9,6), we can solve for a
(6)^6 = 4a (9)
a = 1

Therefore, the distance of the focus from the vertex is 1 m.

We are given with the depth of the telescope of 9 meters and width of the instrument of 12 meter. The shape of the telescope is a parabola. The parabolic equation is y ^2 = 4 a x. When y is 9 and x is equal to 6, then a is 27/ 8 meters or 3.375 meters. The length of the focus thus is 3.375 meters.