How do you write a standard form equation for the hyperbola with  y^2+2y-2x+5=0?

Apr 6, 2016

The given equation is a parabola. It is not a hyperbola.

Explanation:

Here is the graph of the given equation.
graph{y^2+2y-2x+5=0 [-8.375, 11.625, -6.36, 3.64]}
This is a parabola with a horizontal axis of symmetry. As such it's general standard form is
$\textcolor{w h i t e}{\text{XXX}} x = a {y}^{2} + b y + c$ with constants $a , b , c$

Rewriting the given equation:
$\textcolor{w h i t e}{\text{XXX}} {y}^{2} + 2 y - 2 x + 5 = 0$

$\textcolor{w h i t e}{\text{XXX}} 2 x = {y}^{2} + 2 y + 5$

$\textcolor{w h i t e}{\text{XXX}} x = \frac{1}{2} {y}^{2} + 1 y + \frac{5}{2}$
$\textcolor{w h i t e}{\text{XXXXXX}}$(standard parabolic form)