Write an equation of the hyperbola with foci (0, ) and (0, 2) and vertices of (0, ) and (0, 1).
looks like your hyperbola has a vertical major axis
where a=1 and c=2
a^2 + b^2 = c^2
1 + b^2 = 4
b^2 = 3
how about
x^2 - (y^2)/3 = -1
To write the equation of a hyperbola, we need some information about its foci and vertices. In this case, we know the foci are at (0, 0) and (0, 2), and the vertices are at (0, 0) and (0, 1).
The general equation of a hyperbola with a vertical transverse axis (like this one) is:
[(y - k)^2 / a^2] - [(x - h)^2 / b^2] = 1
where (h, k) represents the center of the hyperbola, and a and b are the distances from the center to the vertices.
In this case, the center of the hyperbola is at (h, k) = (0, 0). The distance from the center to the vertices is a = 1.
We also know that the distance between the foci is given by the equation c^2 = a^2 + b^2, where c is the distance from the center to the foci. So, plugging in the values we have, we can solve for b:
c^2 = a^2 + b^2
2^2 = 1^2 + b^2
4 = 1 + b^2
b^2 = 4 - 1
b^2 = 3
b = √3
Now, we can substitute the values of (h, k), a, and b into the equation of the hyperbola:
[(y - 0)^2 / 1^2] - [(x - 0)^2 / (√3)^2] = 1
y^2 - x^2/3 = 1
So, the equation of the hyperbola with the given foci and vertices is y^2 - x^2/3 = 1.