What type of conic section has the equation #9y^2 - x^2 - 4x+54y +68= 0#?
How do I know? Just a quick check of the coefficients on the
1) if the coefficients are both the same number and the same sign, the figure will be a circle.
2) if the coefficients are different numbers but the same sign, the figure will be an ellipse.
3) if the coefficients are of opposites signs, the graph will be a hyperbola.
Let's "solve" it:
Notice that I factored out the leading coefficients already, and gathered together the terms that both have the same variable.
In this step, I completed the square by adding 4 and 9 inside of the parentheses, but then added to the other side, those numbers multiplied by the factored out numbers -1 and 9.
That is what I wanted to see; I can tell what the center of the hyperbola is (-2,-3), how far to move from center to get to vertices (up and down 1 unit since the y-term is divided by 1) and the slope of the asymptotes (