Question 867483
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2.  *[tex \LARGE x^2\ +\ 10x\ -\ 4y^2\ +\ 24y\ =\ 7]


Complete the square on both *[tex \Large x] and *[tex \Large y]


*[tex \LARGE \ \ \ x^2\ +\ 10x\ +\ 25\ -\ 4\left(y^2\ -\ 6y\ +\ 36\right)\ =\ 7\ +\ 25\ -\ 144]


Factor the trinomials


*[tex \LARGE \ \ \ \left(x\ +\ 5\right)^2\ -\ 4\left(y\ -\ 6\right)^2\ =\ -112]


Divide by the constant term to get 1 in the RHS and put the positive term in front.


*[tex \LARGE \ \ \ \frac{\left(y\ -\ 6\right)^2}{28}\ -\ \frac{\left(x\ +\ 5\right)^2}{112}\ =\ 1]


Hyperbola, center at (-5,6), foci and vertices on the line *[tex \Large y\ =\ 6], *[tex \Large a\ =\ \sqrt{28}], *[tex \Large b\ =\ \sqrt{112}]


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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