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] \end{figure}

1. 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 .
2. 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).
3. 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.)
4. 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? \section*{OCR
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