Question

If the telescope is 1 m deep and 8 m wide, how far is the focus from the vertex?

Answer 1

First, let me introduce the general equation of the parabola:

(x-h)^2 = +/- 4a(y-k) or (y-k)^2=+/- 4a(x-h), where

(h,k) are the coordinates of the vertex
a is the distance of the vertex to the focus
4a = length of lactus rectum or the focal width

If the equation contains (x-h)^2, then the parabola passes the x-axis twice. Similarly, (y-k)^2 passes the y-axis twice. If the sign is (-), it opens to the left(if y-axis) or downward (if x-axis). If the sign is (+), it opens to the right(if y-axis) or upward (if x-axis). 

The equation of the parabola is -1/12 x^2 = y. Rearranging to the general form:

x^2 = -12y
Therefore, 

-4a = -12
4a = 12
a = 3, and the parabola is facing downwards.

The vertex is (0,0) at the origin.
The focus is (0,-3). Since it is negative, the focus is situated downwards, hence -3.
The directrix is the mirror image of the focus. Hence, it is a line passing +3 on the y-axis. y=3
Focal width is 4a which is equal to 12 units.