Problem source

Sketch on the same axes the two curves $C_1$ and $C_2$, given by 
\begin{center} 
\begin{align*}
C_1: &&   x y & =  1 \\   
C_2: &&  x^2-y^2 & =   2  
\end{align*}
\end{center} 
The curves 
intersect at $P$ and $Q$. Given that   
 the coordinates of $P$ are $(a,b)$ (which you need not evaluate), 
write down the coordinates of $Q$ in terms of $a$ and $b$. 
 
The tangent to $C_1$ through  $P$ meets the tangent to $C_2$ 
through $Q$ at the point $M$, and the tangent to $C_2$ through $P$ meets the  
tangent to $C_1$ through $Q$ at $N$. Show that the coordinates of $M$ are 
$(-b,a)$ 
and write down the coordinates of $N$. 
 
Show that $PMQN$ is a square.

Solution source

\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){(#1)/sqrt((#1)^2-2*(#1)+1)};
    \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}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\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
    }
    
    % 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, blue, smooth, domain=0.1:\xu, samples=100] 
            plot ({\x}, {1/\x});
        \draw[thick, blue, smooth, domain=\xl:-0.1, samples=100] 
            plot ({\x}, {1/\x});


        \draw[thick, red, smooth, domain=\yl:\yu, samples=100] 
            plot ({sqrt(\x*\x+2)}, {\x});
        \draw[thick, red, smooth, domain=\yl:\yu, samples=100] 
            plot ({-sqrt(\x*\x+2)}, {\x});

    \end{scope}

    \filldraw (0,0) circle (1.5pt) node[above left] {$(0,0)$};
    % \filldraw (2,{\functionf(2)}) circle (1.5pt) node[above] {$(2,\sqrt{2})$};

    % \node[left] at (0,1) {$1$};
    % \node[left] at (0,-1) {$-1$};
    % \node[below] at (0,1) {$1$};
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};

    
    \end{tikzpicture}
\end{center}

$Q = (-a,-b)$

\begin{align*}
&& \frac{\d y}{\d x} &= -\frac{1}{x^2} \\
\Rightarrow && \frac{y-b}{x-a} &= -\frac{1}{a^2} \\
\Rightarrow && 0 &= a^2y+x-a^2b-a \\
&&&= a^2y+x - 2a\\
\\
&& 2x - 2y \frac{\d y}{\d x} &= 0 \\
\Rightarrow && \frac{\d y}{\d x} &= \frac{x}{y} \\
\Rightarrow && \frac{y+b}{x+a} &= \frac{a}{b} \\
\Rightarrow && 0 &= by-ax+b^2 - a^2 \\
&&&= by - ax -2
\end{align*}

Notice that $(-b,a)$ is on both lines, therefore it is their point of intersection. The coordinates of $N$ will be $(a,-b)$.

We can see this is a square by noting each point is a rotation (centre the origin) of $90^\circ$ of each other.