Find an equation in standard form for the hyperbola that satisfies the given conditions.
Center at (0,0) a=4, e=2, horizontal focal axis
clearly the equation is
x^2/a^2 - y^2/b^2 = 1
a=4
e = c/a = 2 so c=8
c^2 = a^2 + b^2
so now you can find b.
h ttps://socratic.org/questions/how-do-you-find-the-standard-form-of-the-hyperbola-that-satisfies-the-given-cond
To find the equation of a hyperbola in standard form given the information provided, we can use the following formula:
(x-h)^2/a^2 - (y-k)^2/b^2 = 1
where (h, k) represents the center of the hyperbola.
Since the center is given as (0, 0), the equation becomes:
(x-0)^2/a^2 - (y-0)^2/b^2 = 1
Simplifying further:
x^2/a^2 - y^2/b^2 = 1
Now, we need to determine the values of a and b. In a hyperbola, the value of a represents the distance from the center to each vertex, and e represents the distance from each focus to the center. Given that a = 4 and e = 2, we can use the relationship between a, b, and e to solve for b:
c = sqrt(a^2 + b^2)
Given that e = 2, we have c = 2. Plugging in the known values:
2 = sqrt(4 + b^2)
Squaring both sides:
4 = 4 + b^2
Subtracting 4 from both sides:
0 = b^2
This implies that b = 0.
Plugging in the values of a = 4 and b = 0 into the equation, we finally get:
x^2/4^2 - y^2/0^2 = 1
Which simplifies to:
x^2/16 - y^2/0 = 1
Simplifying it further, the equation in standard form for the hyperbola is:
x^2/16 - y^2 = 1
To find the equation of the hyperbola in standard form, we need to determine the values of a, b, and c, and then use these values to write the equation.
Given:
- Center at (0,0)
- a = 4
- e = 2
- Horizontal focal axis
The general equation of a hyperbola in standard form with the center at the origin is:
x²/a² - y²/b² = 1
Where a represents the distance from the center to each vertex, and b represents the distance from the center to each co-vertex.
1. First, we need to find the value of b using the relation e² = a² + b².
Since e = 2 and a = 4, we can substitute these values and solve for b²:
2² = 4² - b²
4 = 16 - b²
b² = 16 - 4
b² = 12
b = sqrt(12) = 2√3
2. Now that we have the values of a and b, we can substitute them into the standard form equation to get the equation of the hyperbola:
x²/a² - y²/b² = 1
Plugging in the values a = 4 and b = 2√3, the equation becomes:
x²/4² - y²/(2√3)² = 1
x²/16 - y²/12 = 1
This is the equation of the hyperbola in standard form with a center at (0,0), a = 4, e = 2, and a horizontal focal axis.