Chapter 10.5, Problem 45AYU

To determine

To find: The graph of the equation without applying the rotation of axis: $x^2-7xy+3y^2-y-10=0$.

Answer to Problem 45AYU

Hyperbola.

Explanation of Solution

Given:

$x^2-7xy+3y^2-y-10=0$

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 $B^2-4AC<0$.

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

Calculation:

Consider the given equation $x^2-7xy+3y^2-y-10=0$.

$A=1, B=-7 \text{and} C=3$;

$B^2-4AC={\left(-7\right)}^2-4*1*3=49-12=37>0$

Therefore, the equation defines a hyperbola.