How do I graph -9x^2+25y^2+36x+150y-36=0 algebraically?

Mar 26, 2015

$- 9 {x}^{2} + 36 x + 25 {y}^{2} + 150 y = 36$
$- 9 \left({x}^{2} - 4 x\right) + 25 \left({y}^{2} + 6 y\right) = 36$
$- 9 \left({x}^{2} - 4 x \textcolor{red}{+ 4}\right) + 25 \left({y}^{2} + 6 y \textcolor{red}{+ 9}\right) = 36 \textcolor{red}{- 36 + 225}$
$- 9 {\left(x - 2\right)}^{2} + 25 {\left(y + 3\right)}^{2} = 225$
$\left(\frac{- 9 {\left(x - 2\right)}^{2}}{225}\right) + \left(\frac{25 {\left(y + 3\right)}^{2}}{225}\right) = 1$

$- {\left(x - 2\right)}^{2} / 25 + {\left(y + 3\right)}^{2} / 9 = 1$

${\left(y + 3\right)}^{2} / 9 - {\left(x - 2\right)}^{2} / 25 = 1$

You now have an equation of a hyperbola with vertex at $\left(2 , - 3\right)$, extending vertically with asymptotes at a slope of $\pm \frac{3}{5.}$

Asymptotes: $y = \pm \frac{3}{5} \left(x - 2\right) - 3$
The graph of the function:
graph{(y+3)^2/9 -(x-2)^2/25 = 1 [-10, 10, -5, 5]}