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

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 2

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.