Write an equation for the hyperbola with center(0,-5); asymptotes y=+_ (19 square root6x)-5 , conjugate axis length of 12 units
(+ or -square root of 19 times x divided by 6 )-5
To write the equation of a hyperbola, we need to use the given information about the center, asymptotes, and conjugate axis length.
The standard equation of a hyperbola with its center at (h, k) is given by:
(((x - h)^2) / a^2) - (((y - k)^2) / b^2) = 1
where a and b represent the distances from the center of the hyperbola to the vertices and foci, respectively.
In this case, the center is (0, -5), which gives us h = 0 and k = -5.
We are also given the asymptotes: y = ± (19√6x) - 5. These equations help us find the values of a and b.
The asymptotes of a hyperbola are represented by the equations:
y = ± (b/a)x + k
Comparing this with the given asymptote equation, we can see that b/a = 19√6. Since we know the value of b (the conjugate axis length) is 12 units, we can solve for a by setting up the following ratio:
b/a = 12/a = 19√6
Cross-multiplying, we get:
12 = 19a√6
Dividing both sides by 19√6, we find:
a ≈ 12 / (19√6)
Now that we have the values of a and b, we can write the equation of the hyperbola:
(((x - h)^2) / a^2) - (((y - k)^2) / b^2) = 1
Substituting the known values:
(((x - 0)^2) / (12 / (19√6))^2) - (((y - (-5))^2) / 12^2) = 1
Simplifying further, we get:
(x^2) / ((12 / (19√6))^2) - ((y + 5)^2) / 144 = 1
So, the equation of the hyperbola is:
(x^2) / ((12 / (19√6))^2) - ((y + 5)^2) / 144 = 1