# How do you identity if the equation x^2+4y^2-11=2(4y-x) is a parabola, circle, ellipse, or hyperbola and how do you graph it?

Dec 4, 2016

The given equation represents ellipses.

#### Explanation:

Watch this general form of an equation of conic section -

$A {x}^{2} + B x + C {y}^{2} + D x + E y + F = 0$

If $A \times C = 1$; It is a circle.

If $A \times C > 0$; It is a ellips.

If $A \times C < 1$; It is a Hyperbola.

If $A$ or $B$ is equal to zero, it is a parabola.

Our equation is -

${x}^{2} + 4 {y}^{2} - 11 = 2 \left(4 y - x\right)$

Let us rewrite it in the know form to identify the equation.

${x}^{2} + 4 {y}^{2} - 11 = 8 y - 2 x$

${x}^{2} + 2 x + 4 {y}^{2} - 8 y - 11 = 0$

$A = 1$

$C = 4$

Since $A \times C$ i.e., $1 \times 4 > 0$

The given equation represents ellipses.