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M408M Learning Module Pages
Main page Chapter 10: Parametric Equations and Polar CoordinatesLearning module LM 10.1: Parametrized Curves:Learning module LM 10.2: Calculus with Parametrized Curves:Learning module LM 10.3: Polar Coordinates:Learning module LM 10.4: Areas and Lengths of Polar Curves:Learning module LM 10.5: Conic Sections:Slicing a coneEllipses Hyperbolas Parabolas and directrices Completing the square Learning module LM 10.6: Conic Sections in Polar Coordinates:Chapter 12: Vectors and the Geometry of SpaceChapter 13: Vector FunctionsChapter 14: Partial DerivativesChapter 15: Multiple Integrals |
Completing the squareThe most general conic section has an equation of the form $$Ax^2 + By^2 + Cxy + Dx + E y + F = 0.$$ So far we have taken $C=D=E=0$, except for parabolas. Here we'll see how to adjust for nonzero values of $D$ and $E$. We'll still assume that $C=0$, which means that our curves will point in the direction of the coordinate axes. $C \ne 0$ describes rotated conic sections.
If $A \ne 0$, then we can always absorb the $Dx$ term by completing the square: \begin{eqnarray*} Ax^2 + Dx & = & A \left (x^2 + \frac{D}{A}x \right) \cr \cr & = & A\left (x + \frac{D}{2A}\right )^2 - \frac{D^2}{4A}\end{eqnarray*} Likewise, if $B \ne 0$ then we can absorb the $Ey$ term: $$\displaystyle{By^2 + Ey = B\left (y+\frac{E}{2B}\right )^2-\frac{E^2}{4B}}.$$ If $B=0$ then we cannot absorb the $Ey$ term, and the equation of a parabola winds up looking like $\displaystyle{y-k = \frac{-A}{E} (x-h)^2}$.
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