Write an equation in standard form for the hyperbola with center (0,0), vertex(4,0) , and focus (8,0).

the major axis is horizontal and

a = 4, b = ? and c = 5
I recognize the 3-4-5 Pythagorean triangle, so b = 3

equation:
x^2/16 - y^2/9 = 1

That isn't one of the choses...

1) x^2/64-x^2/16=1
2)x^2/16-y^2/64=1
3)y^2/48-x^2/16=1
4)x^2/16-y^2/48=1

To write the equation of a hyperbola in standard form, we need to know the coordinates of the center, vertices, and foci. For a hyperbola with a horizontal transverse axis (where the center is at (h, k)), the standard form equation is:

(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

In this case, since the center is at (0, 0), and the vertex is at (4, 0), we can determine a value for the "a" term. The distance between the center and vertex (in the horizontal direction) is equal to "a". Thus, a = 4.

Now, we need to calculate the value of "b". In a hyperbola, the distance between the center and each focus is given by "c". We can calculate "c" using the coordinates of the foci.

The distance between the center and focus is 8 units. However, since the center is at (0, 0), the coordinates of the foci are (8, 0) and (-8, 0). Thus, c = 8.

Next, we can calculate the value of "b" using the equation:

c^2 = a^2 + b^2

8^2 = 4^2 + b^2

64 = 16 + b^2

b^2 = 64 - 16

b^2 = 48

Now that we have the values of "a" and "b", we can write the equation in standard form:

(x - 0)^2 / 4^2 - (y - 0)^2 / (√48)^2 = 1

Simplifying, we get:

x^2 / 16 - y^2 / 48 = 1

Therefore, the equation of the hyperbola in standard form with center (0, 0), vertex (4, 0), and focus (8, 0) is:

x^2 / 16 - y^2 / 48 = 1.

To write the equation of the hyperbola in standard form, we need to gather the necessary information: the center, the vertex, and the focus.

Given:
Center: (0,0)
Vertex: (4,0)
Focus: (8,0)

First, we can determine the value of "a" using the distance between the center and vertex. In this case, the distance is 4 units, so "a" is equal to 4.

Next, we can determine the distance between the center and focus, which is the value of "c." In this case, c is equal to 8 units.

We can use the relationship between "a," "b," and "c" to find "b" using the formula c^2 = a^2 + b^2. Plugging in the values, we get:

8^2 = 4^2 + b^2
64 = 16 + b^2
b^2 = 48

To write the equation in standard form for a hyperbola with a horizontal transverse axis, the equation is:

(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

Substituting the given values:
(h,k) = (0,0)
a = 4
b = (sqrt)48

The equation becomes:

x^2 / 16 - y^2 / 48 = 1

Thus, the equation in standard form for the hyperbola with the given center, vertex, and focus is x^2 / 16 - y^2 / 48 = 1.