Find the two integers that have a sum of -6 and a product of 8.
2 answers:
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With the provided vertices and the focus point, the hyperbola will look like the one in the picture below.
notice the "c" distance from the center to the focus point.
since it's a vertical hyperbola, the positive fraction will be the one with the "y" in it, its center is clearly half-way between the vertices at -5, 5.
its major axis or traverse axis goes from -2 up to 12, so is 14 units long, therefore the "a" component is half that, or 7.
![\bf \begin{cases} h=-5\\ k=5\\ a=7\\ c=25 \end{cases}\implies \cfrac{(y-5)^2}{7^2}-\cfrac{[x-(-5)]^2}{b}=1 \\\\\\ \cfrac{(y-5)^2}{7^2}-\cfrac{(x+5)^2}{b}=1\\\\ -------------------------------\\\\ c^2=a^2+b^2\implies \sqrt{c^2-a^2}=b\implies \sqrt{25^2-7^2}=b \\\\\\ \sqrt{625-49}=b\implies \sqrt{576}=b\implies 24=b\\\\ -------------------------------\\\\ \cfrac{(y-5)^2}{7^2}-\cfrac{(x+5)^2}{24^2}=1\implies \cfrac{(y-5)^2}{49}-\cfrac{(x+5)^2}{576}=1](https://tex.z-dn.net/?f=%5Cbf%20%5Cbegin%7Bcases%7D%0Ah%3D-5%5C%5C%0Ak%3D5%5C%5C%0Aa%3D7%5C%5C%0Ac%3D25%0A%5Cend%7Bcases%7D%5Cimplies%20%5Ccfrac%7B%28y-5%29%5E2%7D%7B7%5E2%7D-%5Ccfrac%7B%5Bx-%28-5%29%5D%5E2%7D%7Bb%7D%3D1%0A%5C%5C%5C%5C%5C%5C%0A%5Ccfrac%7B%28y-5%29%5E2%7D%7B7%5E2%7D-%5Ccfrac%7B%28x%2B5%29%5E2%7D%7Bb%7D%3D1%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0Ac%5E2%3Da%5E2%2Bb%5E2%5Cimplies%20%5Csqrt%7Bc%5E2-a%5E2%7D%3Db%5Cimplies%20%5Csqrt%7B25%5E2-7%5E2%7D%3Db%0A%5C%5C%5C%5C%5C%5C%0A%5Csqrt%7B625-49%7D%3Db%5Cimplies%20%5Csqrt%7B576%7D%3Db%5Cimplies%2024%3Db%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0A%5Ccfrac%7B%28y-5%29%5E2%7D%7B7%5E2%7D-%5Ccfrac%7B%28x%2B5%29%5E2%7D%7B24%5E2%7D%3D1%5Cimplies%20%5Ccfrac%7B%28y-5%29%5E2%7D%7B49%7D-%5Ccfrac%7B%28x%2B5%29%5E2%7D%7B576%7D%3D1)
notice the "c" distance from the center to the focus point.
since it's a vertical hyperbola, the positive fraction will be the one with the "y" in it, its center is clearly half-way between the vertices at -5, 5.
its major axis or traverse axis goes from -2 up to 12, so is 14 units long, therefore the "a" component is half that, or 7.