Question 1148688: Given the equation of the parabola: 𝑦^2−8𝑥−4𝑦−20=0. The length of its latus rectum is:
Answer by greenestamps(13200)   (Show Source):
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Put the equation in vertex form,
or
When you have done that, p is the directed distance from the directrix to the vertex, and it is the directed distance from the vertex to the focus. Finally, 4p is the length of the latus rectum.
 given form
 isolate the y terms
 complete the square in y
 put in vertex form
 done. vertex (-3,2); 4p=8
The length of the latus rectum is 8.
In more detail....
The vertex is (-3,2)
4p=8, so p=2.
Since the y term is squared and p is positive, the parabola opens to the right.
p=2 is the directed distance from the directrix to the vertex, so the directrix is 2 units to the left of the vertex, at x=-5.
p=2 is also the directed distance from the vertex to the focus, so the focus is 2 units to the right of the vertex, at (-1,2).
4p=8 is the length of the latus rectum, so the two endpoints of the latus rectum are 2p=4 units above and below the focus, at (-1,6) and (-1,-2).
A graph....
vertex (-3,2)
focus (-1,2)
endpoints of latus rectum: (-1,-2) and (-1,6)
length of latus rectum: 8
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