Answer:
a) Focus is (1,0)
b) Length of Latus rectum is 4
c) The endpoints of the latus rectum is (1,-2) and (1,2)
Step-by-step explanation:
The given parabola has equation
[tex] {y}^{2} = 4x[/tex]
When we compare to
[tex] {y}^{2} = 4px[/tex]
We have
[tex]4px = 4x[/tex]
This implies,
[tex]p = 1[/tex]
The focus of this parabola is at (p,0).
Therefore the focus is (1,0)
b) The length of the latus rectum is given by
[tex] |4p| [/tex]
From a) part we found p to be 1.
Substitute p=1 to obtain:
[tex] |4 \times 1| = 4[/tex]
c) The focus is the midpoint of the latus rectum.
Since the focus is (1,0), we substitute x=1 into the equation to see intersection of the latus rectum and the parabola.
[tex] {y}^{2} = 4(1)[/tex]
This means that:
[tex] {y}^{2} = 4[/tex]
Take square root:
[tex]y = \pm \sqrt{4} [/tex]
[tex]y = \pm2[/tex]
[tex]y = - 2 \: or \: y = 2[/tex]
Therefore the end of latus rectum are at (1,-2) and (1,2)
