Problem source

The line passing through the point $(a,0)$ with gradient $b$ intersects the circle of unit radius centred at the origin at $P$ and $Q$, and $M$ is the midpoint of the chord $PQ$.
Find the coordinates of $M$ in terms of $a$ and $b$.
\begin{questionparts}
 
\item Suppose $b$ is fixed and positive. As $a$ varies, $M$ traces out a curve (the \textit{locus} of $M$). Show that $x=- by$ on this curve.
Given that  $a$ varies with $-1\le a \le 1$, show that the locus is a line segment of length $2b/(1+b^2)^\frac12$.
Give a sketch showing the locus and the unit circle. 
\item Find the locus of $M$ in the following cases, giving in each case its cartesian equation, describing it geometrically and sketching it in relation to the unit circle:
\begin{itemize}
\item $a$ is fixed with $0 < a < 1$, and $b$ varies with  $-\infty < b < \infty$;
\item $ab=1$, and $b$  varies  with $0< b\le1$.
\end{itemize}
\end{questionparts}

Solution source

\begin{align*}
&& y &= bx-ba \\
&& 1 &= x^2 + y^2 \\
\Rightarrow && 1 &= x^2 + b^2(x-a)^2 \\
\Rightarrow && 0 &= (1+b^2)x^2-2ab^2x+b^2a^2-1
\end{align*}

This will have roots which sum to $\frac{2ab^2}{1+b^2}$, therefore $M = \left ( \frac{ab^2}{1+b^2}, \frac{ab^3}{1+b^2}-ba \right)=\left ( \frac{ab^2}{1+b^2}, \frac{-ba}{1+b^2} \right)$



\begin{questionparts}

\item Since $b$ is fixed so is $\frac{b}{1+b^2} = t$ and all the points are $(bta, -ta)$, ie $x = -by$. If $a \in [-1,1]$ we are ranging on the points $(bt, -t)$ to $(-bt, t)$ which is a distance of \begin{align*}
&& d &= \sqrt{(bt+bt)^2+(-2t)^2} \\
&&&= \sqrt{4(b^2+1)t^2} \\
&&&=2 \sqrt{(b^2+1)\frac{b^2}{(b^2+1)^2}} \\
&&&= \frac{2b}{\sqrt{b^2+1}}
\end{align*}


\begin{center}
    \begin{tikzpicture}
    \def\a{2};
    \def\b{0.5};
    \def\t{(\b)/(\b*\b+1)};
    \def\functionf(#1){2*(#1)*((#1)^2-5)/((#1)^2-4)};
    \def\xl{-1.5};
    \def\xu{1.5};
    \def\yl{-1.5};
    \def\yu{1.5};
    
    % 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 (0,0) circle (1);

        \draw[red, thin, dashed] ({-\b*\yl}, {\yl}) -- ({-\b*\yu}, {\yu});

        \draw[blue, thick] ({\b*\t}, {-\t}) -- ({-\b*\t}, {\t});
        
    \end{scope}
    
    % 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}

\item \begin{itemize}
\item If $a$ is fixed we have $\left ( \frac{ab^2}{1+b^2}, -\frac{ba}{1+b^2} \right)$ 
\begin{align*}
&& \frac{x}{y} &= - b \\
\Rightarrow && y &= \frac{a\frac{x}{y}}{1 + \frac{x^2}{y^2}} \\
\Rightarrow && y^2 \left ( 1 + \frac{x^2}{y^2} \right) &= ax \\
\Rightarrow && x^2-ax + y^2 &= 0 \\
\Rightarrow && \left (x - \frac{a}{2} \right)^2 + y^2 &= \frac{a^2}{4}
\end{align*}

Therefore we will end up with a circle centre $(\tfrac{a}{2}, 0)$ going through the origin.


\begin{center}
    \begin{tikzpicture}
    \def\a{2};
    \def\b{0.5};
    \def\t{(\b)/(\b*\b+1)};
    \def\functionf(#1){2*(#1)*((#1)^2-5)/((#1)^2-4)};
    \def\xl{-1.5};
    \def\xu{1.5};
    \def\yl{-1.5};
    \def\yu{1.5};
    
    % 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 (0,0) circle (1);

        % \draw[red, thin, dashed] ({-\b*\yl}, {\yl}) -- ({-\b*\yu}, {\yu});

        \draw[thick, blue, smooth] (0.25, 0) circle (0.25);
        \filldraw (0.5, 0) circle (1.5pt) node[below right]  {$a$};
        % \draw[blue, thick] ({\b*\t}, {-\t}) -- ({-\b*\t}, {\t});
        
    \end{scope}
    
    % 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}

\item If $ab = 1$, we have  $\left ( \frac{b}{1+b^2}, -\frac{1}{1+b^2} \right)$ 

\begin{align*}
&& \frac{x}{y} &= -b \\
\Rightarrow && y &= -\frac{1}{1+\frac{x^2}{y^2}} \\
\Rightarrow && y + \frac{x^2}{y} &= - 1 \\
\Rightarrow && y^2 +y+ x^2 &= 0 \\
\Rightarrow && \left ( y + \frac12 \right)^2 + x^2 &= \frac14
\end{align*}


\begin{center}
    \begin{tikzpicture}
    \def\a{2};
    \def\b{0.5};
    \def\t{(\b)/(\b*\b+1)};
    \def\functionf(#1){2*(#1)*((#1)^2-5)/((#1)^2-4)};
    \def\xl{-1.5};
    \def\xu{1.5};
    \def\yl{-1.5};
    \def\yu{1.5};
    
    % 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 (0,0) circle (1);

        % \draw[red, thin, dashed] ({-\b*\yl}, {\yl}) -- ({-\b*\yu}, {\yu});

        % \filldraw (0.5, 0) circle (1.5pt) node[below right]  {$a$};
        % \draw[blue, thick] ({\b*\t}, {-\t}) -- ({-\b*\t}, {\t});
        
    \end{scope}

    \begin{scope}
        \clip (0, \yl) rectangle (\xu, 0);
        \draw[thick, blue, smooth] (0, -0.5) circle (0.5);
        
    \end{scope}
    
    % 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}

\end{itemize}

\end{questionparts}