Using the given information write the equation of parabolas:

Question

a). the parabola has vertex at the origin and directrix x = −5.

b). the parabola has focus (3, 0), and directrix y=3

PLEASE HELP AND THANK YOU IF YOU DO!!

Answer 1

In order to write a parabola from a focus and a directrix, the one idea to hold onto is that a parabola is the set of points that are the same distance from the focus as they are from the directrix. Everything else follows from that. The vertex sits exactly halfway between the focus and the directrix, and the parabola always opens AWAY from the directrix and toward the focus.

For part a, the vertex is at the origin and the directrix is the vertical line x = -5. Since the directrix is vertical, the parabola opens left or right, and since it opens away from a directrix sitting on the left, it opens to the right. The form we want is:

y^2 = 4px

where p is the distance from the vertex to the focus, which is the same as the distance from the vertex to the directrix. The origin is 5 units from the line x = -5, so p = 5:

y^2 = 4(5)x = 20x

For part b, the focus is (3, 0) and the directrix is the horizontal line y = 3. This time the directrix is horizontal, so the parabola opens up or down, and since the focus at y = 0 sits BELOW the directrix at y = 3, it opens downward.

Remember, the vertex is halfway between the focus and the directrix. Both sit on the vertical line x = 3, and halfway between y = 0 and y = 3 is y = 1.5, so the vertex is (3, 1.5). The distance from the vertex to the focus is p = 1.5.

For a parabola opening down with vertex (h, k), the form is:

(x - h)^2 = -4p(y - k)

Filling in h = 3, k = 1.5 and p = 1.5:

(x - 3)^2 = -6(y - 1.5)

Let us check it. The focus should be p units below the vertex, at (3, 1.5 - 1.5) = (3, 0), and the directrix should be p units above, at y = 1.5 + 1.5 = 3. Both match what we were given.

So, our final answers are y^2 = 20x for part a, and (x - 3)^2 = -6(y - 1.5) for part b.

Answer 2

The standard formula for parabola are the following:

(x - h)² = 4p (y - k) (open up and down)

(y - k)² = 4p (x - h) (opens right or left)

a) based on the equation of the directrix, this one opens to the right. Based on the center at the origin, (h,k) the vertex is (0,0) and p=5. Plugging all this in:

(y - k)² = 4p (x - h)

(y-0)²=4(5)(x-0) Simplify

y²=20x

b) Based on the focus and directrix given, the parabola opens downward. The vertex is in between them, so (h,k) is (3,3/2). p=3/2

(x - h)² = 4p (y - k)

(x-3)²=4(3/2)(y-3/2) Simplify

(x-3)²=6(y-3/2)

Answer 3

Explanation of parabolas and standard formulas:

https://www.mathsisfun.com/geometry/parabola.html

a) If the vertex is at (0,0), and the directrix is at x = -5, then you have a parabola that opens to the right, with the focus at (5,0) (The distance from directrix to the origin is the same as from origin to focus.)

The equation format is y² = 4ax

b) this one is up-facing, and has an offset in the x-axis. The equation will be in the form:

(x - 5)² = 4ay

For either case, the article explains how to find a.

For the second problem, you are given the location of the directrix and the focus. The origin is halfway in between.