Question

A parabola can be drawn given a focus of(-5,−9) and a directrix of

y=−5. Write the equation of the parabola in any form.

Answer 1

To find the equation of the parabola with focus at (-5, -9) and directrix y = -5, we identify the vertex at (-5, -7) and use the standard form equation (x + 5)^2 = 8(y + 7), which can also be expressed in vertex form as y = -1/8(x + 5)^2 - 7.

Step-by-step explanation:

To write the equation of a parabola given a focus at (-5, -9) and a directrix of y = -5, we first note that the vertex of the parabola will be midway between the focus and directrix. In this case, because the focus and directrix are vertically aligned, the vertex will also be on the line x = -5, and its y-coordinate will be the average of -9 (focus y-coordinate) and -5 (directrix y-coordinate), which is -7. Therefore, the vertex is at (-5, -7). Because the directrix is horizontal, this will be a vertically oriented parabola (opening up or down).

Since the distance from the vertex to the focus is 2 units (from -7 to -9), and the same must be true from the vertex to the directrix (from -7 to -5), the focal length f is 2. This information gives us the value 4p = 4f, where p is the distance from the vertex to the focus or directrix. Therefore, 4p = 8.

Now, with the vertex (-5, -7) and 4p = 8, the standard form for the equation of a vertically oriented parabola with a vertex at (h, k) is (x - h)^2 = 4p(y - k). Plugging in the values for h, k, and p, we get (x + 5)^2 = 8(y + 7).

If we wanted to write this parabola equation in vertex form, it would be y = a(x - h)^2 + k, where a affects the width and direction of the parabola. Since the focus is below the directrix, the parabola opens downwards, making a negative in this case. Thus, we have y = -1/8(x + 5)^2 - 7 as the equation of our parabola in vertex form.

Answer 2

`y=-8(x+5)^2-7`

Step-by-step explanation:

Given:

Focus: (-5,-9)

Directrix: y = -5

Find: equation of the parabola

The equation of the line perpendicular to the directrix and passing through the focus is x = -5. This is the line of parabola's symmetry.

The distance between the focus and the directrix is the parabola's parameter, so

`p=|-9-(-5)|=|-9+5|=|-4|=4`

The vertex of the parabola lies on the line of symmetry and divides the distance between the focus and the directrix into two equal parts. So, its coordinates are (-5,-7).

Parabola goes in negative y-direction, thus, the equation of the parabola is

`y-(-7)=-2\cdot 4(x-(-5))^2\\ \\y=-8(x+5)^2-7`