# How do you know if x^2+2y^2+2x-12y+11=0 is a hyperbola, parabola, circle or ellipse?

Jul 19, 2018

Equation of the ellipse ${\left(x + 1\right)}^{2} / {\left(2 \sqrt{2}\right)}^{2} + {\left(y - 3\right)}^{2} / {2}^{2} = 1$

#### Explanation:

${x}^{2} + 2 {y}^{2} + 2 x - 12 y + 11 = 0$

The general equation of conics is

$a {x}^{2} + b x y + c {y}^{2} + d x + e y + f = 0$ , for ellipse or circle

${b}^{2} - 4 a c < 0$ for parabola , ${b}^{2} - 4 a c = 0$ and hyperbola,

b^2-4 ac > 0 ; a= 1, b= 0 , c=2 :. b^2-4a c <0 :.circle or

ellipse. ${\left(x + 1\right)}^{2} + 2 {\left(y - 3\right)}^{2} = 19 - 11$ or

${\left(x + 1\right)}^{2} + 2 {\left(y - 3\right)}^{2} = 8$ or

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

This is standard equation of horizontal ellipse is

 x^2/a^2+y^2/b^2=1 ; a>0 , b>0

Hence this is ellipse.

graph{x^2+2y^2+2x-12y+11=0 [-20, 20, -10, 10]} [Ans]