Question

Determine equation of the Porabola that opens right, has vertex of (-10,-4), and has a focal diameter of 20

Answer 1

The equation of the parabola that opens to the right with vertex (-10, -4) and a focal diameter of 20 is (y + 4)^2 = 40x + 400.

Step-by-step explanation:

To find the equation of a parabola that opens to the right with a vertex at (-10, -4) and a focal diameter of 20, we first need to determine the focus and the directrix of the parabola. The focal length (distance between the vertex and the focus) is half the focal diameter, so it is 20/2 = 10. This implies that the focus is 10 units to the right of the vertex, at the point (0, -4).

The general form of a parabola opening to the right is (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p is the distance from the vertex to the focus. Substituting the given vertex and focal length, we get (y + 4)^2 = 40(x + 10), which simplifies to (y + 4)^2 = 40x + 400 as the equation of the parabola.

Answer 2

The equation of the parabola that opens right, with a vertex at (-10, -4), and a focal diameter of 20 is
`\((x + 10)^2 = 20(y + 4)\).`

The equation of the parabola that opens right, has a vertex of (-10, -4), and has a focal diameter of 20.

Since the parabola opens to the right, its equation will be of the form:

`(x - h)^2 = 4a(y - k)`

where (h, k) is the vertex of the parabola, and a is the distance between the vertex and the focus.

Given that the vertex is (-10, -4) and the focal diameter is 20, we can find the value of a as follows:

focal diameter = 4a

20 = 4a

a = 5

Now that we know the value of a, we can plug it into the equation to get:

`(x - (-10))^2 = 4 * 5(y - (-4))`

Simplifying the equation, we get:

`(x + 10)^2 = 20(y + 4)`

Therefore, the equation of the parabola is:

`(x + 10)^2 = 20(y + 4)`