What is an equation of the parabola with vertex at the origin and focus (-4,0)

A. x=1/16y2
B. x=-1/16y2
C. y=-1/16x2
d. y=1/16x2

To determine the equation of a parabola given its vertex and focus, we can use the geometric definition of a parabola.

The geometric definition of a parabola states that the distance from any point on the parabola to the focus is equal to the perpendicular distance from that point to the directrix.

In this case, the vertex of the parabola is at the origin (0, 0), and the focus is (-4, 0). By using the geometric definition, we can determine the equation of the parabola.

First, let's determine the distance from the origin to the focus (-4, 0). The distance formula is given by:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using the coordinates of the origin as (0, 0) and the focus as (-4, 0), the distance formula becomes:

d = sqrt((-4 - 0)^2 + (0 - 0)^2)
= sqrt((-4)^2 + 0)
= sqrt(16)
= 4

The distance from any point on the parabola to the focus is 4.

Now, let's determine the perpendicular distance from any point on the parabola to the directrix. Since the vertex is at the origin, the equation of the directrix is of the form x = a, where a is the distance from the vertex to the directrix. Since the vertex is at the origin, the directrix is a vertical line passing through x = -a.

In this case, the vertex is at the origin, so the equation of the directrix is x = -a.

The perpendicular distance from a point (x, y) on the parabola to the directrix x = -a is given by |x + a|. Therefore, in this case, the perpendicular distance from a point on the parabola to the directrix is |x|.

According to the geometric definition of a parabola, the distance from the point on the parabola to the focus (4) should be equal to the perpendicular distance from the point to the directrix |x|.

Therefore, the equation of the parabola can be written as:

| x | = 4

To get rid of the absolute value, we can square both sides of the equation:

x^2 = 16

Hence, the equation of the parabola is x^2 = 16.

Therefore, the correct answer to the question is not provided in the given options A, B, C, and D.

x^2 = 4py

where p = -4

So, ...