Step By Step Calculus » 3.1 - General Conics

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Synopsis
Double Napped Cone and Conic Sections The conic sections are produced when a plane intersects a double napped cone, two cones attached at a vertex. The vertical line through the vertex is called the central axis, considered to have an angle 0^{\circ}0^{\circ}, and the diagonal sides are called generators, considered to have an angle \theta\theta measured clockwise from the central axis. The following table summarizes the conic sections based on the angle \phi\phi between the intersecting plane and the central axis.
\begin{tabular}{l|l|c}\hline Type & Conic & Angle, $\phi$\\\hline & Parabola & $\theta$\\ Non-degenerate conic sections & Circle & $90^{\circ}$\\ (plane does not pass through vertex) & Ellipse & $\theta<\phi<90^{\circ}$\\ & Hyperbola & $0\le\phi<\theta$\\ \hline & Single line & $\theta$\\ Degenerate conic sections &A point & $\theta<\phi\le90^{\circ}$\\ (plane passes through vertex) &Two intersecting lines & $0\le\phi<\theta$\\ & & \\\hline \end{tabular}
General and Standard Forms of Conic Sections The general form that represents all conics is given by
\displaystyle \textbf{(General Form)} \qquad Ax^2+Bxy+Cy^2+Dx+Ey+F=0.
\displaystyle \textbf{(General Form)} \qquad Ax^2+Bxy+Cy^2+Dx+Ey+F=0.
In our course we will only deal with the case B=0B=0. When B\ne 0B\ne 0 the axis of the cone is not parallel to either of the xx or yy axes. So for simplicity in this first course we will deal only with the case when the cone has its axis parallel to either the xx or yy axes. The different conics, parabola, circle, ellipse and hyperbola are generated by different conditions on the coefficients A,B,C,D,E,A,B,C,D,E, and FF. Again we can make things simple by dealing with a special version of the standard form (completing the square when B=0B=0, AA and CC are not zero) that allows us to easily identify the conditions which give us the different conics.
\displaystyle \textbf{(Intermediate Form)} \quad \frac{(x-h)^2}{c}+\frac{(y-k)^2}{d}=1\quad\begin{array}{l}\textrm{with } h=-\frac{D}{2A},k=-\frac{E}{2C},\\ c=-\frac{F}{A}+\frac{D^{2}}{4A^{2}}+\frac{E^{2}}{4AC},d=-\frac{F}{C}+\frac{D^{2}}{4AC}+\frac{E^{2}}{4C^{2}}\end{array}
\displaystyle \textbf{(Intermediate Form)} \quad \frac{(x-h)^2}{c}+\frac{(y-k)^2}{d}=1\quad\begin{array}{l}\textrm{with } h=-\frac{D}{2A},k=-\frac{E}{2C},\\ c=-\frac{F}{A}+\frac{D^{2}}{4A^{2}}+\frac{E^{2}}{4AC},d=-\frac{F}{C}+\frac{D^{2}}{4AC}+\frac{E^{2}}{4C^{2}}\end{array}
As we work through the details for each type of conic, we will see how this intermediate standard form varies.