Problems

Prove that the diagonal triangle of four coplanar points is self-polar with respect to any conic through the four points. \(A, B, C, D\) are four points on a rectangular hyperbola and the chords \(BC, AD\) are perpendicular and meet at \(X\); prove that the polar line of \(X\) with respect to the rectangular hyperbola is the radical axis of the circles which have \(BC, AD\) as diameters.

(i) Find the locus of the feet of the perpendiculars from a fixed point \(O\) to the straight lines which pass through another fixed point and lie in a plane which does not pass through the fixed point \(O\). (ii) Prove that there are two real points \(P, Q\) in space at which each of the sides of an acute angled triangle \(ABC\) subtends a right angle. Prove also that, if \(D\) is any point on \(PQ\), the opposite edges of the tetrahedron \(ABCD\) are perpendicular to each other. \item[] N.B. The equations in Questions 6, 7, 8, 9 are referred to rectangular Cartesian coordinate axes.

From a variable point \(P\) of the line \(p \equiv ax+by+c=0\) a perpendicular \(PL\) is drawn to the line \(p' \equiv a'x+b'y+c'=0\), and a point \(Q\) is taken on \(PL\), such that \(\vec{PQ} = k.\vec{QL}\); prove that the locus of \(Q\) is the line \(q \equiv (a^2+b^2)p+k(aa'+bb')p'=0\). Verify that, when \(k=-2\), the line \(p'\) is one of the angle bisectors of the lines \(p, q\), and state the geometrical reason for this.

Find the condition that the circle through the points \((ka^2, ka), (kb^2, kb), (kc^2, kc)\) should pass through the point \((0,0)\). A circle through the vertex of the parabola \(y^2-kx=0\) cuts the parabola again in the points \(A, B, C\); prove that the centroid of the triangle formed by the tangents at \(A, B, C\) lies on the axis of the parabola.

State (without proof) how the conic, whose envelope (tangential) equation is \[ (x_1l+y_1m+1)(x_2l+y_2m+1) - \kappa^2(l^2+m^2)=0, \] is related to the points \((x_1, y_1), (x_2, y_2)\) and the length \(\kappa\). The parabolas given by the envelope equations \begin{align*} (al+bm)(x_1l+y_1m+1)+\lambda(l^2+m^2)&=0, \\ (bl-am)(x_2l+y_2m+1)+\mu(l^2+m^2)&=0, \end{align*} have three common tangents other than the line at infinity; prove that the equation of the circumcircle of the triangle formed by these tangents is \[ (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0. \]

Prove that the normals to the conic \(ax^2+2hxy+by^2+c=0\) at its intersections with the rectangular hyperbola \((hx+by)(x-x_0)-(ax+hy)(y-y_0)=0\), pass through the point \((x_0, y_0)\). Explain geometrically why, for all values of \((x_0, y_0)\), this rectangular hyperbola has its asymptotes parallel to the lines \(h(x^2-y^2)-(a-b)xy=0\).

Prove that the equations of the sides of a quadrilateral may, by a suitable choice of the triangle of reference, be expressed in any system of homogeneous coordinates \((x,y,z)\) in the forms \(lx \pm my \pm nz = 0\). If the equation of the line at infinity is \(ax+by+cz=0\), prove that the middle points of the three diagonals of the quadrilateral lie on the line \(\frac{l^2x}{a} + \frac{m^2y}{b} + \frac{n^2z}{c}=0\).

If \(n\) is any positive integer, shew that \(n\) consecutive odd integers can be found not one of which is prime. Shew also that one such sequence has sum \[ \frac{1}{2}\left\{ 2n+2 - \sqrt{2n+1} + n(n+2) \right\}. \]

Define a recurring series, its scale of relation, and generating function. Shew that the series whose \(n\)th term is \(2^n n^2 x^n\) is a recurring series and find its generating function. For what range of values of \(x\) does the latter represent the sum to infinity of the series?

If \(f(x)\) and \(\phi(x)\) are two polynomials in \(x\), explain and justify a general method of finding their highest common factor. Find the highest common factor of \[ 6x^6+6x^5+13x^4+15x^3-2x^2-5x-12 \] and \[ 4x^7+4x^6+x^5+5x^4-16x^3-23x^2-4x-6. \]