Problems

\(Q, R\) are two points on a rectangular hyperbola subtending a right angle at a point \(P\) of the curve. Prove that \(QR\) is parallel to the normal at \(P\).

If \(l_1 = 0, l_2 = 0\) are the equations of two lines, and if \(S = 0\) is the equation of a conic, interpret the equation \(S + \lambda l_1 l_2 = 0\), where \(\lambda\) is a parameter. Hence, or otherwise, show that if a circle meets an ellipse in four points, the joins of these points in pairs are equally inclined to the axis of the ellipse. Circles are drawn to meet the ellipse \(x^2/a^2 + y^2/b^2 = 1\) in four points lying in pairs on two lines through the fixed point \((x_1, y_1)\). Show that the circles form a co-axial system, and find the equation of the radical axis.

Let \(A(\theta)\) and \(B(\theta)\) denote the matrices $$\begin{pmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{pmatrix}, \quad \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$ respectively.

1. [(i)] If \(\begin{pmatrix} X' \\ Y' \end{pmatrix} = A(\theta) \begin{pmatrix} X \\ Y \end{pmatrix}\), show that the point \((X', Y')\) is the mirror image of \((X, Y)\) in the line \(y = x\tan\frac{1}{2}\theta\).
2. [(ii)] Prove that \(A(\theta_1) A(\theta_2) = B(\phi)\), where \(\phi\) is some angle (to be determined), and hence, or otherwise, explain the relation between the points \((X', Y')\) and \((X, Y)\) when \(\begin{pmatrix} X' \\ Y' \end{pmatrix} = B(\theta) \begin{pmatrix} X \\ Y \end{pmatrix}\).
3. [(iii)] Prove that \(A(\theta_1) A(\theta_2) A(\theta_3) = A(\theta_4) A(\theta_5) A(\theta_1)\), and interpret this result geometrically.

\(S\) is the set of real numbers. Operations, denoted by \(\oplus\) and \(\otimes\), are defined on \(S\) by \begin{align} a \oplus b &= a + b + 1,\\ a \otimes b &= ab + a + b, \end{align} where the operations of addition and multiplication on the right are the usual ones. Show that if \(\oplus\) and \(\otimes\) are taken to define an addition and multiplication on \(S\), then \(S\) is a field. Which element of \(S\) has no multiplicative inverse in this field, and what is the multiplicative inverse of a general element \(x\) of \(S\)?