write any equation of a hyperbola with a vertical transverse axis
Sure! A hyperbola with a vertical transverse axis has the general equation:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
In this equation, (h, k) represents the coordinates of the center of the hyperbola. The values "a" and "b" represent the distance from the center to the vertices along the x-axis and y-axis, respectively.
To obtain the specific equation of a particular hyperbola, you'll need additional information such as the coordinates of the center and the lengths of the axes. The coordinates of the center are necessary to determine the values of (h, k), while the lengths of the axes will help determine the values of "a" and "b".
If you have the coordinates of the center, let's say (h, k), and the distances from the center to the vertices, let's call them "c", you can find the values of "a" and "b" using the relation a^2 = c^2 - b^2.
With these values, you can substitute them into the general equation to obtain the equation of the hyperbola.