Posted by Anonymous on Monday, March 24, 2014 at 9:45am.
Which equation represents a hyperbola whose foci lie at (2, 7) and (2, 3)?

Math  PreCalc (12th Grade)  Steve, Monday, March 24, 2014 at 11:17am
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
(y2)^2/9  (x2)^2/16 = 1
Verify this at
http://www.wolframalpha.com/input/?i=plane+curve+%28y2%29^2%2F9++%28x2%29^2%2F16+%3D+1

Math  PreCalc (12th Grade)  Steve, Monday, March 24, 2014 at 11:18am
or, to see the foci plotted,
http://www.wolframalpha.com/input/?i=foci+of+%28y2%29^2%2F9++%28x2%29^2%2F16+%3D+1
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