refreshConicSections.tex

%\documentclass{ximera}
%
%\input{../preamble.tex}
%
%\title[Refresh:]{Conic sections and their algebraic descriptions}
%
%\begin{document}
%\begin{abstract}
%  Conic sections are examples of famous curves in mathematics.
%\end{abstract}
%\maketitle
%
%\section{Conics Sections}
%The \dfn{conic sections} are circles, ellipses, parabolas, and
%hyperbolas.  In this section you'll explore the origin of the name
%``conic sections.''
%
%The interactive figure shows a cone, formed by rotating a line in
%space around some fixed axis, and a plane.  Use the sliders for
%$\alpha$ and $\beta$ to produce each of the four conic sections:
%
%If the embedded interactive below does not display correctly, click \link[here]{https://www.geogebra.org/m/xHXwN8wK}
%
%\geogebra{xHXwN8wK}{1366}{480}
%
%\begin{problem}
%  Use the figure to help you select the true statements:
%  \begin{selectAll}
%    \choice{Any plane perpendicular to the cone's axis produces a circle.}
%    \choice[correct]{Any plane parallel to the cone's axis and doesn't pass through the vertex produces an hyperbola.}
%    \choice[correct]{A plane skew to the cone that doesn't pass through the vertex produces an ellipse.}
%    \choice[correct]{A plane can intersect a cone in one point.}
%  \end{selectAll}
%\end{problem}
%
%\section{Algebraic description of conic sections}
%The general algebraic description of all the conic sections in cartesian coordinates is given by
%\[
%  Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,
%\]
%for constants $A$, $B$, $C$, $D$, $E$, and $F$.
%For most of our cases, the conic sections will be symmetric with respect to either the $x$- or $y$-axes.
%
%Therefore we can take $B = 0$ in the above equation and further
%simplifications are possible.
%
%You can find the standard forms of circles, ellipses, and hyperbolas online. For instance, check \link[here]{https://www.varsitytutors.com/hotmath/hotmath_help/topics/conic-sections-and-standard-forms-of-equations}
%
%\begin{problem}
%  What is the standard from of an ellipse centered at the origin with
%  vertices at $(\pm a,0)$ and foci at $(\pm c ,0)$?
%  \begin{explanation}
%    The standard form of an ellipse is
%    \[
%    \frac{x^2}{a^2} + \frac{y^2}{b^2}=1\qquad c^2 = a^2 - b^2
%    \]
%  \end{explanation}
%\end{problem}
%
%
%
%\begin{problem}
%  What is the standard from of a hyperbola centered at the origin with
%  vertices at $(0,\pm a)$ and foci at $(0,\pm c)$?
%  \begin{explanation}
%    The standard form of an ellipse is
%    \[
%    \frac{y^2}{a^2} - \frac{x^2}{b^2}=1\qquad c^2 = a^2 + b^2
%    \]
%  \end{explanation}
%\end{problem}
%
%
%\begin{problem}
%  Consider
%  \[
%  y^2 = -36x
%  \]
%  This is the formula for a
%  \wordChoice{\choice{ellipse}\choice{hyperbola}\choice[correct]{parabola}}.
%  Its focus is at $\left(-9,0\right)$ and its directrix is the line $x
%  = \answer{9}$. 
%\end{problem}
%
%
%\end{document}