SOLUTION: Find the vertex, focus, directrix, and focal width of the parabola. x = 10y2 choices are below Vertex: (0, 0); Focus: one divided by forty comma zero ; Directrix: x = negative

Algebra ->  Trigonometry-basics -> SOLUTION: Find the vertex, focus, directrix, and focal width of the parabola. x = 10y2 choices are below Vertex: (0, 0); Focus: one divided by forty comma zero ; Directrix: x = negative       Log On


   

Question 1135977: Find the vertex, focus, directrix, and focal width of the parabola.
x = 10y2
choices are below
Vertex: (0, 0); Focus: one divided by forty comma zero ; Directrix: x = negative one divided by forty ; Focal width: 0.1

Vertex: (0, 0); Focus: one divided by ten comma zero ; Directrix: x = negative one divided by ten ; Focal width: 0.1

Vertex: (0, 0); Focus: one divided by forty comma zero ; Directrix: x = one divided by forty ; Focal width: 40

Vertex: (0, 0); Focus: zero comma one divided by forty ; Directrix: y = negative one divided by forty ; Focal width: 40
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

4p%28x-h%29=%28y-k%29%5E2 is the standard equation for a right-left facing parabola with vertex at (h, k )
rewrite x=10y%5E2 in the standard form :
x=10y%5E2.........both sides divide by 10
x%2F10=10y%5E2%2F10
x%2F10=y%5E2
factor 4
4%28%281%2F10%29%2F4%29x=y%5E2....simplify
4%281%2F40%29x=y%5E2
rewrite as
4%281%2F40%29%28x-0%29=%28y-0%29%5E2
so, (h,+k )= (0, 0 ), p=+1%2F40+
parabola is symmetric around the x-axis and so the focus lies a distance p from the center (0, 0) along the x-axis
(0%2Bp,0)
(0%2B1%2F40,0)
(1%2F40,0)->focus
the distance between the focus and directrix is p=+1%2F40+
parabola is symmetric around the x-axis and so the directrix is a line parallel to the y-axis, a distance -p from the center left (0,0) x-coordinate
x=0-p
x=0-1%2F40
x=-1%2F40
so, you have:
vertex at (0, 0 ),
focus at (1%2F40,0)
directrix is x=-1%2F40

you can use only this one of your choices as an answer
Vertex: (0, 0);
Focus: (1%2F40,0);
Directrix: x = 1/40; -> this is incorrect, should be highlight%28x=-1%2F40%29
Focal width: 40



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


For me, the most useful form of the equation of a parabola (that opens right or left) is

x-h+=+%281%2F%284p%29%29%28y-k%29%5E2

In this form...
(1) the vertex is (h,k);
(2) p is the (directed) distance from the vertex to the focus; which means -p is the directed distance from the vertex to the directrix; and
(3) 4p is the focal width (length of the latus rectum)

Written in that form, the equation in your example is

x-0+=+%281%2F%281%2F10%29%29%28y-0%29%5E2

So...
(1) the vertex is (0,0);
(2) 4p=1/10 so p=1/40, so the focus is (1/40,0) and the directrix is x = -1/40; and
(3) the focal width is 4p = 1/10 = 0.1

The first answer choice is the correct one.