Question
Prove that the graph of the equation $$A x^{2}+C y^{2}+D x+E y+F=0$$ is one of the following (except in degenerate cases).Conic(a) Circle(b) Parabola(c) Ellipse(d) Hyperbola Condition$A=C$$A=0$ or $C=0$ (but not both)$A C>0$$A C<0$
Step 1
This is a general second-degree equation, which can represent any conic section depending on the values of the coefficients \( A, C, D, E, \) and \( F \). Show more…
Show all steps
Your feedback will help us improve your experience
Ankit Gupta and 68 other Calculus 2 / BC educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Prove that the graph of the equation $A x^{2}+C y^{2}+D x+E y+F=0$ is one of the following (except in degenerate cases). (a) Circle (b) Parabola (c) Ellipse $A=c$ $A=0$ or $C=0$ (but not both) $A C>0$ $\Delta C<0$
Topics in Analytic Geometry
Hyperbolas
Proof Prove that the graph of the equation $A x^{2}+C y^{2}+D x+E y+F=0$ is one of the following (except in degenerate cases). $\begin{array}{ll}{\text { Conic }} & {\text { Condition }} \\ {\text { (a) Circle }} & {A=C} \\ {\text { (b) Parabola }} & {A=0 \text { or } C=0 \text { (but not both) }} \\ {\text { (c) Ellipse }} & {A C > 0} \\ {\text { (d) Hyperbola }} & {A C < 0}\end{array}$
Conics, Parametric Equations, and Polar Coordinates
Conics and Calculus
Show that the graph of an equation of the form $$ A x^{2}+C y^{2}+D x+E y+F=0 \quad A \neq 0, C \neq 0 $$ where $A$ and $C$ are opposite in sign, (a) is a hyperbola if $\frac{D^{2}}{4 A}+\frac{E^{2}}{4 C}-F \neq 0$ (b) is two intersecting lines if $\frac{D^{2}}{4 A}+\frac{E^{2}}{4 C}-F=0$
Analytic Geometry
The Hyperbola
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD