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
failed

Solution

failed
failed

Solution Steps

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

When A=CA = C, the equation Ax2+Cy2+Dx+Ey+F=0A x^{2} + C y^{2} + D x + E y + F = 0 simplifies to A(x2+y2)+Dx+Ey+F=0A(x^{2} + y^{2}) + D x + E y + F = 0. This represents a circle because the coefficients of x2x^{2} and y2y^{2} are equal and nonzero.

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

When AC=0A C = 0, either A=0A = 0 or C=0C = 0. If A=0A = 0, the equation becomes Cy2+Dx+Ey+F=0C y^{2} + D x + E y + F = 0, which is a parabola. Similarly, if C=0C = 0, the equation becomes Ax2+Dx+Ey+F=0A x^{2} + D x + E y + F = 0, which is also a parabola. Thus, when AC=0A C = 0, the conic is a parabola.

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

When ACA \neq C and AC>0A C > 0, the coefficients of x2x^{2} and y2y^{2} are both nonzero and have the same sign. This indicates that the conic is an ellipse. If A=CA = C, it would be a circle, but since ACA \neq C, it is an ellipse.

Final Answer

If A=CA = C, then the answer is a circle\boxed{\text{a circle}}.

If AC=0A C = 0, then the answer is a parabola\boxed{\text{a parabola}}.

If ACA \neq C and AC>0A C > 0, then the answer is an ellipse\boxed{\text{an ellipse}}.

If AC<0A C < 0, then the answer is a hyperbola\boxed{\text{a hyperbola}}.

Was this solution helpful?
failed
Unhelpful
failed
Helpful