Chapter 10.5, Problem 47AYU

To determine

To find: The graph of the equation without applying the rotation of axis: $9x^2+12xy+4y^2-x-y=0$.

Answer to Problem 47AYU

Parabola.

Explanation of Solution

Given:

$9x^2+12xy+4y^2-x-y=0$

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 $B^2-4AC<0$.

c. Defines a hyperbola if $B^2-4AC>0$.

Calculation:

Consider the given equation $9x^2+12xy+4y^2-x-y=0$.

$A=9, B=12 \text{and} C=4$;

$B^2-4AC={\left(12\right)}^2-4*9*4=144-144=0$

Therefore, the equation defines a parabola.