In this article (and video above), we tackle a conic sections problem step by step, helping you determine the center of the curve and identify its type. By the end of this article, you’ll have the tools and confidence to handle similar questions with ease.
Question:
In this question, we are presented with the general equation of a conic section, and our job can be split up into two objectives: First, we need to determine the center of the conic section and secondly, we must identify the type of conic section it represents. Before diving into the solution, let’s take a moment to review some essential concepts that will help us approach this problem effectively.
Determine the center of the conic section described by the following general equation, and identify the type of conic section it represents:
9𝑥2 + 16𝑦2 − 54𝑥 + 64𝑦 = 311
Problem Context: Conic Sections
We start off by looking at what ‘conic sections’ refers to. This can be defined as any curve formed by the intersection of a plane with a right circular cone, as illustrated here. The type of curve—whether it’s an ellipse, parabola, circle, or hyperbola—depends on the angle at which the plane intersects the cone. Mathematically, we can distinguish between the different curves created by this intersecting plane using two key angles.
The first angle, denoted as θ (theta) in the FE Handbook, represents the angle between the intersecting plane and the vertical axis of the cone. The second angle, φ (phi), is the vertex angle, measured between the vertical axis of the cone, and its slanted outer surface. If these two angles are known, we can calculate what is referred to as the eccentricity of the conic sections using the formula:
[Read more…] about Mastering Conic Sections Made EASY for the FE Exam!
