Chapter 10.5, Problem 6AYU

Except for degenerate cases, the equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$ defines a(n) _______ if $B^2-4AC=0$.

(a) circle

(b) ellipse

(c) hyperbola

(d) parabola

To determine

To fill: Except for degenerate cases, the equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$ defines a(n) _______ if $B^2-4AC=0$.

a. circle

b. ellipse

c. hyperbola

d. parabola

Answer to Problem 6AYU

d. parabola.

Explanation of Solution

Given:

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

Formula used:

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

a. Defines a parabola if $B²=4AC$.

b. Defines an ellipse $\left(A\ne C\right)$ or a circle $\left(A=C\right)$ if $B²-4AC< 0$.

c. Defines a hyperbola if $B²-4AC> 0$.

Calculation:

For the equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$;

If $B²-4AC=0$, then the equation defines a parabola.