Questions: A nondegenerate conic section in the form Ax^2 + Cy^2 + Dx + Ey + F = 0, in which A and C are not both zero is Answer if A=C. Answer if AC=0, Answer if A ≠ C and AC>0, Answer if AC<0 - a hyperbola - a circle - a parabola - an ellipse

A nondegenerate conic section in the form Ax^2 + Cy^2 + Dx + Ey + F = 0, in which A and C are not both zero is

Answer if A=C.

Answer if AC=0,

Answer if A ≠ C and AC>0,

Answer if AC<0

- a hyperbola
- a circle
- a parabola
- an ellipse

Transcript text: A nondegenerate conic section in the form $A x^{2}+C y^{2}+D x+E y+F=0$, in which A and C are not both zero is Answer if $A=C$. Answer if $A C=0$, Answer if $A \neq C$ and $A C>0$, Answer if $A C<0$ :: a hyperbola :: a circle :: a parabola :: an ellipse

Solution

Solution Steps

Step 1: Analyze the case when $A = C$

When $A = C$, the equation $A x^{2} + C y^{2} + D x + E y + F = 0$ simplifies to $A(x^{2} + y^{2}) + D x + E y + F = 0$. This represents a circle because the coefficients of $x^{2}$ and $y^{2}$ are equal and nonzero.

Step 2: Analyze the case when $A C = 0$

When $A C = 0$, either $A = 0$ or $C = 0$. If $A = 0$, the equation becomes $C y^{2} + D x + E y + F = 0$, which is a parabola. Similarly, if $C = 0$, the equation becomes $A x^{2} + D x + E y + F = 0$, which is also a parabola. Thus, when $A C = 0$, the conic is a parabola.

Step 3: Analyze the case when $A \neq C$ and $A C > 0$

When $A \neq C$ and $A C > 0$, the coefficients of $x^{2}$ and $y^{2}$ are both nonzero and have the same sign. This indicates that the conic is an ellipse. If $A = C$, it would be a circle, but since $A \neq C$, it is an ellipse.