determine if the hyperbola is vertical or horizontal
then find the vertices foci and asymptotes
y^2/25-x^2/100=1
My answers
Horizontal?
no,
multiply by -1
x^2/100 - y^2/25 = -1
the left side starts with +x^2
and the right side is -1
thus vertical
Proof:
http://www.wolframalpha.com/input/?i=y%5E2%2F25-x%5E2%2F100%3D1
To determine if the hyperbola is vertical or horizontal, we can examine the equation of the hyperbola.
In the given equation:
y^2/25 - x^2/100 = 1
Notice that the x^2 term is being subtracted from the y^2 term. This indicates that the hyperbola is horizontal. If the x^2 term was being subtracted from the y^2 term, then it would be a vertical hyperbola.
So, given that the equation shows subtraction of x^2 from y^2, we can conclude that the hyperbola is horizontal.
Now let's find the vertices, foci, and asymptotes of the hyperbola.
The general equation for a hyperbola in standard form is:
((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1
Comparing our equation to the general equation, we can determine the values of a, b, h, and k.
For our equation:
a^2 = 100, so a = 10
b^2 = 25, so b = 5
The center of the hyperbola is at the point (h, k). In our equation, the center is (0, 0) since there are no values subtracted from x^2 or y^2. So, h = 0 and k = 0.
Using this information, we can find the vertices, foci, and asymptotes:
The vertices are determined by moving a units horizontally from the center (h, k) and b units vertically. The vertices for a horizontal hyperbola are at (h ± a, k). In our case, the vertices are (0 ± 10, 0) which gives us (-10, 0) and (10, 0).
The foci are determined by moving c units horizontally from the center (h, k). The foci for a horizontal hyperbola are at (h ± c, k). The value of c can be found using the equation c^2 = a^2 + b^2. In our case, c^2 = 100 + 25 = 125, so c = √125 ≈ 11.18. The foci are (0 ± 11.18, 0), which gives us (-11.18, 0) and (11.18, 0).
The asymptotes of a hyperbola pass through the center (h, k) and have slopes equal to ± (b / a). In our case, the asymptotes have slopes ± (5 / 10) = ± 0.5 (since b = 5 and a = 10). Since the center is (0, 0), the equations of the asymptotes are y = 0.5x and y = -0.5x.
Therefore, the answer is:
The hyperbola is horizontal.
The vertices are (-10,0) and (10,0).
The foci are (-11.18,0) and (11.18,0).
The asymptotes are y = 0.5x and y = -0.5x.