Chapter 10.5, Problem 56AYU

To determine

To prove: Except for degenerate cases $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$.

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

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

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

Answer to Problem 56AYU

a. The equation represents a parabola.

b. The equation represents an ellipse.

c. The equation represents a hyperbola.

Explanation of Solution

Given:

The Equations $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$.

Proof:

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

$B^2-4AC=B^{\text{'}2}-4A\text{'}C\text{'}$ for any rotation $\theta$. Let $\theta =45$, So that $B^{\prime }=0$.

$B^2-4AC=-4A\text{'}C\text{'}$

a. If $B^2-4AC=-4A\text{'}C\text{'}\text{ = 0}$ then, $A\text{'}C\text{'}=0$, Using the theorem for identifying conics without completing the square, the equation represents a parabola.

b. If $B^2-4AC=-4A\text{'}C\text{'}<0$ then, $A^{\prime }{C}^{\prime }>0$ thus the equation represents an ellipse.

c. If $B^2-4AC=-4A^{\prime }{C}^{\prime }>0$ then, $A^{\prime }{C}^{\prime }<0$ thus the equation represents a hyperbola.