What conic is $x^2-y^2-2y-2=0$ and what is the conic format of it? It would be great if work was shown so I could learn it.

Question

What conic is $x^2 - y^2 - 2y - 2=0$ and what is the conic format of it? It would be great if work was shown so I could learn it. I know it's either a hyperbola or an eclipse but I don't know how to convert it to one

Answer 1

$x^2 -(y+1)^2 = 1$, canyou tell what it is?

Answer 2

Hint

When you are so lucky that the equation does not contain the unpleasant $x\times y$ term and is written $$Ax^2+By^2+Cx+Dy+E=0$$ just complete the squares to get $$A\Big(x+\frac C{2A}\Big)^2+B\Big(y+\frac D{2B}\Big)^2+\Big(E-\frac{C^2}{4 A}-\frac{D^2}{4 B}\Big)=0$$ and you easily see what the conic is : if $AB>0$ it is an ellipse ( if moreover $A=B$ it is a circle), if $AB<0$ it is an hyperbola, if $AB=0$ it is a parabola.