a drink bottler needs to fill 20 one-pint bottles, and he has two tanks: one that contains 2 gallons of soda, and a second that
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Check the picture below.
so, the hyperbola looks like so, clearly a = 6 from the traverse axis, and the "c" distance from the center to a focus has to be from -3±c, as aforementioned above, the tell-tale is that part, therefore, we can see that c = 2√(10).
because the hyperbola opens vertically, the fraction with the positive sign will be the one with the "y" in it, like you see it in the picture, so without further adieu,
![\bf \begin{cases} h=2\\ k=-3\\ a=6\\ c=2\sqrt{10} \end{cases}\implies \cfrac{[y- (-3)]^2}{ 6^2}-\cfrac{(x- 2)^2}{ b^2}=1 \\\\\\ \cfrac{(y+3)^2}{ 36}-\cfrac{(x- 2)^2}{ b^2}=1 \\\\\\ c^2=a^2+b^2\implies (2\sqrt{10})^2=6^2+b^2\implies 2^2(\sqrt{10})^2=36+b^2 \\\\\\ 4(10)=36+b^2\implies 40=36+b^2\implies 4=b^2 \\\\\\ \sqrt{4}=b\implies 2=b\\\\ -------------------------------\\\\ \cfrac{(y+3)^2}{ 36}-\cfrac{(x- 2)^2}{ 2^2}=1\implies \cfrac{(y+3)^2}{ 36}-\cfrac{(x- 2)^2}{ 4}=1](https://tex.z-dn.net/?f=%5Cbf%20%5Cbegin%7Bcases%7D%0Ah%3D2%5C%5C%0Ak%3D-3%5C%5C%0Aa%3D6%5C%5C%0Ac%3D2%5Csqrt%7B10%7D%0A%5Cend%7Bcases%7D%5Cimplies%20%5Ccfrac%7B%5By-%20%28-3%29%5D%5E2%7D%7B%206%5E2%7D-%5Ccfrac%7B%28x-%202%29%5E2%7D%7B%20b%5E2%7D%3D1%0A%5C%5C%5C%5C%5C%5C%0A%5Ccfrac%7B%28y%2B3%29%5E2%7D%7B%2036%7D-%5Ccfrac%7B%28x-%202%29%5E2%7D%7B%20b%5E2%7D%3D1%0A%5C%5C%5C%5C%5C%5C%0Ac%5E2%3Da%5E2%2Bb%5E2%5Cimplies%20%282%5Csqrt%7B10%7D%29%5E2%3D6%5E2%2Bb%5E2%5Cimplies%202%5E2%28%5Csqrt%7B10%7D%29%5E2%3D36%2Bb%5E2%0A%5C%5C%5C%5C%5C%5C%0A4%2810%29%3D36%2Bb%5E2%5Cimplies%2040%3D36%2Bb%5E2%5Cimplies%204%3Db%5E2%0A%5C%5C%5C%5C%5C%5C%0A%5Csqrt%7B4%7D%3Db%5Cimplies%202%3Db%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0A%5Ccfrac%7B%28y%2B3%29%5E2%7D%7B%2036%7D-%5Ccfrac%7B%28x-%202%29%5E2%7D%7B%202%5E2%7D%3D1%5Cimplies%20%5Ccfrac%7B%28y%2B3%29%5E2%7D%7B%2036%7D-%5Ccfrac%7B%28x-%202%29%5E2%7D%7B%204%7D%3D1)
so, the hyperbola looks like so, clearly a = 6 from the traverse axis, and the "c" distance from the center to a focus has to be from -3±c, as aforementioned above, the tell-tale is that part, therefore, we can see that c = 2√(10).
because the hyperbola opens vertically, the fraction with the positive sign will be the one with the "y" in it, like you see it in the picture, so without further adieu,