5 Fig. 5 shows a circle with centre \(\mathrm { C } ( a , 0 )\) and radius \(a\). B is the point \(( 0,1 )\). The line BC intersects the circle at P and \(\mathrm { Q } ; \mathrm { P }\) is above the \(x\)-axis and Q is below.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66ca36f1-099c-44ce-a6e2-027172e44fd8-4_556_659_539_742}
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\caption{Fig. 5}
\end{figure}
- Show that, in the case \(a = 1 , \mathrm { P }\) has coordinates \(\left( 1 - \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)\). Write down the coordinates of Q .
- Show that, for all positive values of \(a\), the coordinates of P are
$$x = a \left( 1 - \frac { a } { \sqrt { a ^ { 2 } + 1 } } \right) , \quad y = \frac { a } { \sqrt { a ^ { 2 } + 1 } } .$$
Write down the coordinates of Q in a similar form.
Now let the variable point P be defined by the parametric equations \(( * )\) for all values of the parameter \(a\), positive, zero and negative. Let Q be defined for all \(a\) by your answer in part (ii).
- Using your calculator, sketch the locus of P as \(a\) varies. State what happens to P as \(a \rightarrow \infty\) and as \(a \rightarrow - \infty\).
Show algebraically that this locus has an asymptote at \(y = - 1\).
On the same axes, sketch, as a dotted line, the locus of Q as \(a\) varies.
(The single curve made up of these two loci and including the point B is called a right strophoid.) - State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle at which a right strophoid crosses itself?
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