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\noindent {\bf Problem statement} In this problem two different
families of curves are considered.  The families are said to be {\it
orthogonal}\/ to one another, since at every point where a curve from
the first family intersects a curve from the second family, the
tangent lines of the two curves are at right angles.

\medskip

\noindent a) Sketch the curves $x^4 -3y^4 = C$ for $C = -1$, $0$, $1$,
and $2$ in the viewing window $[.5,2] \times [.25,1.25]$.  Also,
sketch the curves $3x^{-2} + y^{-2} = D$ for $D = 3$, $4$, $5$, and
$6$ in the same window.
 
\smallskip

\noindent {\bf Suggestion:} First solve to find $y$ explicitly as a
function of $x$, remembering that $y > 0$ in the region given.

\medskip

\noindent b) Use calculus to verify that the two families are
orthogonal.

\smallskip

\noindent {\bf Suggestion:} Use implicit differentiation on the
equation that defines each family of curves and express $\displaystyle
{{dy}\over{dx}}$ as a function of $x$ and $y$.

%contributed by Michael O'Nan

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