Problems

Prove that the feet of the four normals from \((\xi, \eta)\) to the ellipse \[ S \equiv x^2/a^2 + y^2/b^2 - 1 = 0 \] lie on the rectangular hyperbola \[ (a^2-b^2)xy + b^2\eta x - a^2\xi y = 0. \] Deduce that the normals to \(S\) at the four points where it is cut by the lines \[ lx/a + my/b - 1 = 0, \quad x/(la) + y/(mb) + 1 = 0 \] are concurrent. \par Hence show that, if the normals at three points \(L, M, N\) of \(S\) meet on \(S\) and \(L\) is the point \((a\cos\theta, b\sin\theta)\), the equation of \(MN\) is \[ x \cos\theta/a^3 - y\sin\theta/b^3 + 1/(a^2-b^2) = 0. \]

A chord \(PQ\) is normal to a rectangular hyperbola \(S\) at \(P\), and another chord \(LM\) is drawn parallel to \(PQ\). Show that \(PL\) and \(QM\) meet on the diameter perpendicular to \(CP\), where \(C\) is the centre of \(S\).

\(POP'\), \(QOQ'\) and \(ROR'\) are three concurrent chords of a conic \(S\), and \(X\) is any other point of \(S\). \(QR, XP'\) meet in \(L\); \(RP, XQ'\) in \(M\); \(PQ, XR'\) in \(N\). Prove that \(L, M, N\) lie on a line through \(O\).

\(A, B, C\) are three fixed points in the plane of a conic \(S\), and \(M\) is a variable point of \(S\). \(AM\) meets \(S\) again in \(N\), and \(BN\) meets \(S\) again in \(L\). Prove that if, for all positions of \(M\) on \(S\), the points \(C, L, M\) are collinear, the triangle \(ABC\) is self-conjugate with respect to \(S\).

The straight line \[ l \equiv \alpha x + \beta y + \gamma z = 0 \] meets the sides \(BC, CA, AB\) of the triangle of reference in \(A_1, B_1, C_1\). \(A'\) is the harmonic conjugate of \(A_1\) with respect to \(B\) and \(C\), and \(B', C'\) are similarly defined. Show that \(AA', BB', CC'\) meet in a point \(O\) and find the equation of the conic \(S\) through \(A, B, C\) for which \(O\) and \(l\) are pole and polar. \par Prove further that, if \(l\) touches the conic \[ x^2+y^2+z^2-2yz-2zx-2xy=0, \] the conic \(S\) passes through the point \((1,1,1)\).

If \(H\) is the orthocentre of a triangle \(ABC\) and if \(AH\) cuts \(BC\) in \(D\) and the circumcircle again in \(X\), prove that \(HD=DX\), with similar results for \(BH\) and \(CH\). Derive from this result another theorem by inverting with centre \(H\).

Prove that, if a variable chord of a circle subtends a right angle at a fixed point, the locus of its pole is a circle.

Two points \(A, B\) in space are on the same side of a plane. Find a point \(P\) in the plane such that the sum of the distances \(PA, PB\) is a minimum.

The base \(BC\) of a triangle is given. Find the locus of the vertex \(A\) when (i) the sum of the base angles \(B, C\) is given; (ii) the difference of the base angles is given.

Two rectangular hyperbolas intersect in \(A, B, C, D\). Prove that all conics through \(A, B, C, D\) are rectangular hyperbolas. \par Deduce that (i) the orthocentre of a triangle inscribed in a rectangular hyperbola lies on the curve; (ii) if a chord of rectangular hyperbola subtends a right angle at a point \(O\) of the curve, then the chord is parallel to the normal at \(O\).