Fig. 5 shows a circle with centre C \((a, 0)\) and radius \(a\). B is the point \((0, 1)\). The line BC intersects the circle at P and Q. P is above the \(x\)-axis and Q is below.
\includegraphics{figure_5}
- Show that, in the case \(a = 1\), P has coordinates \(\left(1 - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\). Write down the coordinates of Q. [3]
- Show that, for all positive values of \(a\), the coordinates of P are
$$x = a\left(1 - \frac{a}{\sqrt{a^2 + 1}}\right), \quad y = \frac{a}{\sqrt{a^2 + 1}} \quad (*)$$
Write down the coordinates of Q in a similar form. [4]
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 \(a \to \infty\) and as \(a \to -\infty\).
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. [8]
(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? [3]