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\centerline{MONT 108N -- Mathematics Through Time}
\centerline{Discussion 4/Problem Set 5 -- Coordinate Geometry}
\centerline{November 22, 2010}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
Euclidean-style geometry, where there is no notion
of numerical values for lengths or areas is often 
called \emph{synthetic} geometry.  The French mathematician
and philosopher \emph{Ren\'e Descartes} (1596 - 1650 C.E.) thought 
that the Euclidean formulation made it too difficult to 
prove results about geometric figures (at least once they got more complicated than
straight lines, polygons, and circles).   His solution was
to introduce numerical {\it coordinates} for the points 
in the plane, which were horizontal and vertical distances 
measured with respect to a pair of {\it coordinate
axes}.  The resulting system of $xy$-coordinates is often called
a {\it Cartesian} coordinate system in his honor.    The idea
is contained in the following picture:
\begin{center}
\begin{picture}(100,120)
\put(60,10){\line(0,1){100}}
\put(-10,50){\line(1,0){140}}
\put(120,40){$x$}
\put(63,110){$y$}
\put(90,70){\circle*{4}}
\put(95,70){$P = (a,b)$}
\put(75,40){$a$}
\put(53,57){$b$}
\put(90,70){\line(0,-1){20}}
\put(90,70){\line(-1,0){30}}
\put(90,53){\line(1,0){3}}
\put(93,53){\line(0,-1){3}}
\put(63,70){\line(0,-1){3}}
\put(63,67){\line(-1,0){3}}
\end{picture}
\end{center}
This idea has become so standard by now that even the typesetting
software I am using to create this document uses (a version of) Descartes' idea to 
create pictures like the one above(!)  For instance, the commands
that produced the picture above look like this in this system 
(called \LaTeX):
\begin{verbatim}
\begin{picture}(100,120)
\put(60,10){\line(0,1){100}}
\put(-10,50){\line(1,0){140}}
\put(120,40){$x$}
\put(63,110){$y$}
\put(90,70){\circle*{4}}
\put(95,70){$P = (a,b)$}
\put(75,40){$a$}
\put(53,57){$b$}
\put(90,70){\line(0,-1){20}}
\put(90,70){\line(-1,0){30}}
\put(90,53){\line(1,0){3}}
\put(93,53){\line(0,-1){3}}
\put(63,70){\line(0,-1){3}}
\put(63,67){\line(-1,0){3}}
\end{picture}
\end{verbatim}
Can you guess what each command does?

Any such way of doing geometry by means of coordinates is 
known as as sort of \emph{analytic geometry}.  (There are other
ways of giving coordinates as well, including the {\it polar}
$(r,\theta)$-coordinates that some of you may know.)  
\vskip 10pt
\noindent
{\it Coordinate Geometry -- Discussion Questions}
\vskip 10pt
One of the features of Descartes' system is that geometric 
figures may now be defined as the sets of solutions of \emph{algebraic
equations}.  This means that Cartesian coordinates let us do \emph{algebraic 
geometry} (as opposed to the Babylonians' supposed \emph{geometric algebra}).  
\vskip 10pt
\noindent
A.  \emph{Straight Lines}
\begin{enumerate}
\item[1.]  What sort of $xy$-coordinate equation is satisfied
by the points of a \emph{straight line} in the plane?  Are there 
special cases not covered by the general form(s)?   What do the various
constants in your equations tell you about the line in geometric terms?
\item[2.]  If $A,B,C$ are constants and not both $A,B$ are zero, then
say why the equation $Ax + By + C = 0$ on the coordinates of a general
point $(x,y)$ is satisfied if and only if $(x,y)$ is on a straight line.
How does this form of equation relate to what you said in part 1?
\end{enumerate}
\vskip 10pt
\noindent
B.  \emph{Circles} -- Following Euclid's Postulate 3, suppose
we construct a circle with center at a general point $O$ with coordinates
$(h,k)$, and the radius has length $r > 0$. 
\begin{enumerate}
\item[1.] What $xy$-equation
satisfied by all the points $(x,y)$ on the circle?  Hint:  Consider the following
figure
\begin{center}
\begin{picture}(100,60)
\put(10,10){\circle*{4}}
\put(-20,-5){$(h,k)$}
\put(10,10){\line(1,2){20}}
\put(30,50){\circle*{4}}
\put(40,55){$(x,y)$}
\put(10,30){$r$} 
\put(10,10){\line(1,0){20}}
\put(30,10){\line(0,1){40}}
\put(27,10){\line(0,1){3}}
\put(27,13){\line(1,0){3}}
\end{picture}
\end{center}
constructed with a line through $(h,k)$ parallel to the $x$-axis
and a line through $(x,y)$ parallel to the $y$-axis (using Euclid's Proposition 
31).  What are the sides in this right triangle?  What relation must hold?
\item[2.]  Consider any equation of the form 
$x^2 + y^2 + Ax + By + C = 0$.  Show that the set of all solutions is
either a circle, a point, or empty, depending on the relative
sizes of the constant coefficients $A,B,C$.  Hint:  complete the square
in $x$ and $y$. 
\end{enumerate}
\vskip 10pt
\noindent
C. \emph{Parabolas} -- A parabola can be defined in geometric
terms as the \emph{locus} (that is, the whole collection) of 
points that are equidistant from a given point (the focus), and
a given line (the {\it directrix}).  
\begin{enumerate}
\item[1.]  For instance, let the focus be the point with coordinates 
$(0,p)$ on the $y$-axis with $p > 0$, and let the directrix be the 
line with equation $y = -p$.  The distance from a point $P = (x,y)$ 
to this line is just the distance along the perpendicular from
$P$ to the directrix.  Draw a picture showing the points that satisfy
the condition defining the parabola.
\item[2.]  Find the coordinate equation of the parabola.  
\end{enumerate}
\vskip 10pt
\noindent
D.  \emph{Ellipses} -- This one is harder, but it illustrates
the true power of the Cartesian coordinate system.  Ancient Greek
mathematicians (especially Apollonius) were able to handle 
conic sections like this by Euclidean methods, but their approach
was among the most complicated and advanced mathematics of the time.
One way to define an ellipse is this: An ellipse is the locus of 
all points $(x,y)$ such that 
\begin{quotation}
\emph{the sum of the distances from $(x,y)$ to
two fixed points (called the foci) is a constant.}
\end{quotation} 
\begin{enumerate}
\item[1.]  Suppose the foci are at the points $(-1,0)$ and $(1,0)$
and the sum of the distance from $(x,y)$ to $(-1,0)$
and the distance from $(x,y)$ to $(1,0)$ is $4$.  Work
out an $xy$-equation for the resulting ellipse that \emph{has no square roots}.
(This will require squaring \emph{twice}, and carefully
collecting and cancelling terms at the end.)  
\item[2.]  You might
recall from high school geometry that the general equation
for an ellipse with foci at points $(\pm c,0)$ 
on the $x$-axis can be put into the form 
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
for some constants $a > b$ (the semi-major and semi-minor axes).
What are those constants for the ellipse you found in part 1,
and what do they tell us geometrically about the ellipse? 
\end{enumerate}
\vskip 10pt
\noindent
{\it Assignment}
\vskip 10pt
You will discuss these questions in the discussion groups in
class on Monday, November 22.  But these problems will be
due {\it as an individual problem set} on Friday, December 3.
\end{document} 
