Similarities and Differences among parabola, ellipse and hyperbola

All the students who would be able to read this article and, coincidently, who are taking Math 51 in MSU-IIT, this is something that you must keep in mind.

One lesson in your course is the conic section. This three-part lesson is too bulky because it is comprised of lessons in parabola, ellipse and hyperbola which all present some varying concepts and properties. You might get confused of what the conic section you are dealing with especially if you have poorly familiarized the properties of these three. For this reason, I have summed up some of the similarities and differences of parabola, ellipse and hyperbola for you to familiarize.

The first difference is with regards to the eccentricity of the conic section. If the eccentricity is equal to one, the graph of the conic section looks like a parabola. However, if the eccentricity is lesser than one and greater than one then you are dealing with ellipse and hyperbola respectively. Second is on the location of the vertex (or vertices) and the focus (or foci) of the conic section on the graph. There is only a focus and a vertex of the graph of the parabola. On the other hand, the graphs of the ellipse and hyperbola would show that there are two vertices and foci, and that they are collinear with the center on a particular axis. The difference between the graph of the ellipse and hyperbola is mainly on the position of their foci and vertices. The foci of an ellipse come next to the center of the conic section at a certain distance. However, the foci of the hyperbola come after the vertices of hyperbola; the vertices of the hyperbola come next to the center at a certain distance. Lastly, the difference between these conic sections can be noticed on the general form of the equation of the conic section. The degree of the polynomial of the parabola, ellipse and the hyperbola is 2. However, only a variable in the equation of the parabola is raised to this certain degree (This is true only to the equations of parabola which graph is on a translated or untranslated axes.) unlike in the equations of both ellipse and hyperbola which variable, say x and y, are both raised to 2. These are just few of the similarities and differences of parabola, ellipse and hyperbola which can help you in dealing with these conic sections. It might help you a lot but remember that in dealing with these conic sections, you actually need not get familiarized with the above properties as long as you understand thoroughly the concepts that vary the conic sections among each other.