Chapter 10.5, Problem 52AYU

In Problems 43-52, identify the graph of each equation without applying a rotation of axes.

$3x^2+2xy+y^2+4x-2y+10=0$

To determine

To find: The graph of the equation without applying the rotation of axis: $3x^2\text{ + }2xy+y^2+4x-2y\text{ + }10=0$.

Answer to Problem 52AYU

Ellipse.

Explanation of Solution

Given:

$3x^2\text{ + }2xy+y^2+4x-2y\text{ + }10=0$

Formula used:

The Equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0,B\ne 0$.

a. Defines a parabola if $B^2-4AC=0$.

b. Defines an ellipse (or Circle) if $B^2-4AC\text{ < }0$.

c. Defines a hyperbola if $B^2-4AC\text{ > }0$.

Calculation:

Consider the given equation $3x^2\text{ + }2xy+y^2+4x-2y\text{ + }10=0$.

$A=3,B=2\text{ and }C=1$;

$B^2-4AC={\left(2\right)}^2-4*3*1=4-12=-8<0$

Therefore, the equation defines an ellipse.