Question

Find coordinates of the focus, the length of the latus rectum, and the coordinates of its endpoint for y^2= 4x

Answer 1

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

${y}^{2} = 4x$

When we compare to

${y}^{2} = 4px$

We have

$4px = 4x$

This implies,

$p = 1$

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

$|4p|$

From a) part we found p to be 1.

Substitute p=1 to obtain:

$|4 \times 1| = 4$

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.

${y}^{2} = 4(1)$

This means that:

${y}^{2} = 4$

Take square root:

$y = \pm \sqrt{4}$

$y = \pm2$

$y = - 2 \: or \: y = 2$

Therefore the end of latus rectum are at (1,-2) and (1,2)