Conic sections and the discovery of a novel curve using differential equations

We began by observing a variety of properties related to the tangent and normal lines of three conic sections: a parabola, an ellipse, and a hyperbola. Some of these properties include specific relationships between the x-and y-intercepts of the tangent and normal lines. Using these properties, we were able to form several differential equations. Afterwards, we took an arbitrary differentiable function satisfying those differential equations and attempted to solve them in order to find whether or not the function represents a conic section. Some of these differential equations were nontrivial, but to our surprise, we were still able to solve a few of them using a combination of standard methods. The main highlight of the paper was the discovery of a brand-new closed curve, which was obtained by mimicking a certain property satisfied by a normal line of a parabola. Another highlight of the paper was the creation of a novel set of unified definitions to describe the three conic sections using differential equations. These results are interesting because the traditional definitions involve the concepts of eccentricity and directrix, or sections of a double cone by a plane, while ours only deals with the x-intercepts of the tangent lines. Mathematica (R) was also used to create several animations that illustrate some of the discoveries in this paper.