Which equation represents a hyperbola whose foci lie at (2, 7) and (2, -3)?

there are lots of them, depending on the eccentricity. Recall that an hyperbola with equation

y^2/a^2 - x^2/b^2 = 1

has vertices at (0,±a) and foci at (0,±c) where c = ae and e>1 is the eccentricity.

Also, recall that c^2 = a^2 + b^2

Here, we have the center of the graph at (2,2) with c=5. So, we can pick a and b as long as a^2+b^2 = 5^2. A happy relation, no? How about a=3,b=4?

So, your hyperbola is shifted some, making its equation

(y-2)^2/9 - (x-2)^2/16 = 1

Verify this at

http://www.wolframalpha.com/input/?i=plane+curve+%28y-2%29^2%2F9+-+%28x-2%29^2%2F16+%3D+1

or, to see the foci plotted,

http://www.wolframalpha.com/input/?i=foci+of+%28y-2%29^2%2F9+-+%28x-2%29^2%2F16+%3D+1

To find the equation of a hyperbola, we need to know the coordinates of the foci and some other information such as the center or the vertices. In this case, we have the coordinates of the foci, but we don't have the center or vertices.

The standard form equation of a hyperbola with horizontal transverse axis is:

((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1

And the standard form equation of a hyperbola with vertical transverse axis is:

((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1

To determine if the transverse axis is horizontal or vertical, we look at the positions of the foci. In this case, the foci have the same x-coordinate but different y-coordinates. Therefore, the transverse axis is vertical.

To find the values for a^2 and b^2, we need additional information such as the center or the vertices. Without that information, we cannot determine the exact equation of the hyperbola.

It's worth mentioning that if the center of the hyperbola is given as (h, k), we could use the distance formula to find the value of a^2 or b^2. The distance between the foci and the center (c) can be found using the formula c^2 = a^2 + b^2.

Without the given center or vertices, we are unable to fully determine the equation of the hyperbola.