\documentclass[11pt]{report}
\input{header}
\pagestyle{empty}
\begin{document}
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    \rput(1.96,-3){%
      \psframebox*{%
        \begin{minipage}[t]{5cm}
          \textcolor{blue}{$c_1: 3x-5y-2=0$}\\
          \textcolor{olivegreen}{$c_2: x^2+3y^2-10=0$}\\
          \textcolor{orange}{$c_3: x^2+y^2+xy^2=0$}\\
          \textcolor{purple}{$c_4: x^6+y^6-x^3y^3-12=0$}
        \end{minipage}
      }
    }
    %
    % C1: 3x-5y-2=0 (line)
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    % C2: x^2+3*y^2-10=0 (ellipse)
    \parametricplot[linecolor=olivegreen,plotstyle=ccurve]{0}{360}{10 sqrt t cos mul 10 3 div sqrt t sin mul}
    % C3: x^2+y^2+xy^2=0 (2 disconnected curves + an isolated point)
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    \psplot[linecolor=orange]{-4}{-1.075}{0 x sub -1 x sub sqrt div}
    \arrColorVertex(0,0){3}{orange}
    % C4: x^6+y^6-x^3y^3-12=0
    \parametricplot[linecolor=purple,plotstyle=ccurve]{0}{360}{
      /sqrttwelve 12 sqrt def
      /third 1 3 div def
      /u sqrttwelve t sin mul 2 t cos mul add def
      /v sqrttwelve t sin mul 2 t cos mul sub def
      /su u abs third exp def
      /sv v abs third exp def
      u 0 ge {su} {su neg} ifelse
      v 0 ge {sv} {sv neg} ifelse
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    \rput(3,-1.25){\textcolor{olivegreen}{$c_2$}}
    \rput(-1.75,2.4){\textcolor{orange}{$c_3$}}
    \rput(1.85,-0.5){\textcolor{purple}{$c_4$}}
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\end{document}
