Write an equation in standard form for the hyperbola with center (0,0), vertex(4,0) , and focus (8,0).
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
From the given information, you know that a = 4 and c = 8.
Using b^2 = c^2 - a^2, we get this:
b^2 = 8^2 - 4^2 = 64 - 16 = 48
Substituting 16 for a^2 and 48 for b^2 gives us this in standard form:
x^2/16 - y^2/48 = 1
I hope this helps.
Thank you again.
To write the equation of a hyperbola in standard form, you need to determine the values of the horizontal and vertical components.
Given that the center is (0,0) and the vertices are (4,0) and (8,0), we can determine that the horizontal distance between the vertices is 8 - 4 = 4.
The standard form of a hyperbola with a horizontal transverse axis is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
where (h,k) is the center and 'a' represents the distance between the center and the vertex along the horizontal transverse axis.
In this case, the center is (0,0) and the horizontal distance between the center and vertex is 4. Therefore, 'a' is equal to 4.
Now, let's substitute the values in the equation:
(x - 0)^2 / 4^2 - (y - 0)^2 / b^2 = 1
Simplifying further:
x^2 / 16 - y^2 / b^2 = 1
Since we are given the focus (8,0), we know that the distance between the focus and the vertex along the horizontal axis is 'c', which is equal to 8 - 4 = 4.
The relationship between 'a', 'b', and 'c' in a hyperbola is defined by the equation:
c^2 = a^2 + b^2
Substituting the values:
4^2 = 16 + b^2
16 = 16 + b^2
b^2 = 0
Since b^2 is equal to 0, this indicates that it is a degenerate case of a hyperbola.
Hence, the correct equation in standard form for the given hyperbola is:
x^2 / 16 - y^2 / 0 = 1
Simplifying it further, we get:
x^2 / 16 = 1
Thus, the correct option is 2) x^2/16 - y^2/64 = 1.