Problem source

The point $P$ has coordinates $(x,y)$ which satisfy
\[
x^2+y^2 + kxy +3x +y =0\,.
\]
\begin{questionparts}
\item Sketch the locus of $P$ in the case $k=0$, giving the points of intersection with the coordinate axes.
\item By factorising $3x^2 +3y^2 +10xy$, or otherwise, sketch the locus of $P$ in the case $k=\frac{10}{3}\,$, giving the points of intersection with the coordinate axes.
\item In the case $k=2$, let $Q$ be the point obtained by rotating $P$ clockwise about the origin by an angle~$\theta$, so that the coordinates $(X,Y)$ of $Q$ are given by 
\[
X=x\cos\theta +y\sin\theta\,, \ \ \  \ Y= -x\sin\theta + y\cos\theta\,.
\]
Show that, for $\theta =45^\circ$, the locus of $Q$  is 
$ \sqrt2 Y=   (\sqrt2 X+1  )^2 - 1 .$ 
Hence, or otherwise, sketch the locus of $P$ in the case $k=2$, giving the equation of the 
line of symmetry.
\end{questionparts}

Solution source

\begin{questionparts}
\item $k = 0$, we have $x^2 + y^2 + 3x + y = 0$, ie $(x+\tfrac32)^2+(y+\tfrac12)^2 = \frac{10}{4}$.

\begin{center}
    \begin{tikzpicture}
        \draw[thick, ->] (-4, 0) -- (4, 0) node [right] {$x$};
        \draw[thick, ->] (0, -4) -- (0, 4) node [above] {$y$};

        \draw[thick, blue, smooth, domain=0:360, samples=100]
            plot({1.5 + sqrt(10)/2*cos(\x)}, {0.5 + sqrt(10)/2*sin(\x)});

        \filldraw (1.5, 0.5) circle (1pt) node[above] {$(\tfrac32,\tfrac12)$};
        \filldraw (0,0) circle (1pt) node[above left] {$(0,0)$};
        \filldraw (3,0) circle (1pt) node[below right] {$(3,0)$};
        \filldraw (0,1) circle (1pt) node[left] {$(0,1)$};
    \end{tikzpicture}
\end{center}

\item $3x^2 + 3y^2 +10xy = (3x+y)(x+3y)$ so $x^2 + y^2 + \tfrac{10}3xy + 3x+y = (3x+y)(\frac{x+3y}{3}+1) = 0$ so we have the line pair $3x +y =0$, $x+3y + 3 = 0$

\begin{center}
    \begin{tikzpicture}
        \draw[thick, ->] (-4, 0) -- (4, 0) node [right] {$x$};
        \draw[thick, ->] (0, -4) -- (0, 4) node [above] {$y$};

        \draw[thick, blue, smooth] ({4/3},-4) -- ({-4/3}, 4);
        \draw[thick, blue, smooth] (-4,{1/3}) -- (4, {-7/3});

        \filldraw (0,0) circle (1pt) node[above left] {$(0,0)$};
        \filldraw (0,-1) circle (1pt) node[left] {$(0,-1)$};
        \filldraw (-3,0) circle (1pt) node[above] {$(0,-3)$};
    \end{tikzpicture}
\end{center}

\item If $k = 2$ then $(x+y)^2 + (x+y)+2x = 0$. If $\theta = 45^\circ$ then $ X = \frac1{\sqrt{2}}(x+y), Y = \frac{1}{\sqrt{2}}(y-x)$, ie $x+y = \sqrt{2}X$ and $x = \frac{1}{\sqrt2}(X-Y)$, so our equation is:
\begin{align*}
0 &= 2X^2 + \sqrt{2}X + \sqrt{2}(X-Y)  \\
&= (\sqrt{2}X + 1)^2 - 1 - \sqrt{2} Y
\end{align*}
which would be a parabola with line of symmetry $X = -\frac{1}{\sqrt{2}}$.

However, we are actually looking at that parabola rotated by $45^\circ$ anticlockwise.

\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){floor(#1)/(#1)^2)};
    \def\xl{-4};
    \def\xu{4};
    \def\yl{-4};
    \def\yu{4};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{1}
    \pgfmathsetmacro{\yscale}{\xrange/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);
        
        \draw[thick, red, dashed, smooth, domain=-4:4, samples=100] 
            plot (\x, {(sqrt(2)*\x+1)^2 -1});

        \draw[thick, blue, smooth, domain=-4:4, samples=100] 
            plot ({\x * cos(45) - ((sqrt(2)*\x+1)^2 -1)*sin(45)}, {\x * sin(45) + ((sqrt(2)*\x+1)^2 -1)*cos(45)});

        \draw[thick, blue, smooth, domain=-4:4, samples=100] 
            plot ({(-1/sqrt(2)) * cos(45) - \x*sin(45)}, {(-1/sqrt(2)) * sin(45) + \x*cos(45)});
        
    \end{scope}

    
    \end{tikzpicture}
\end{center}

\end{questionparts}