Given that \[xy - 3x - 2y + 4 = 0,\] evaluate \[\frac{(x-1)(y-4)}{(x-4)(y-1)}.\] If also \[xz - 6x - z + 8 = 0,\] find numbers \(p\), \(q\) such that \[\frac{(y-p)(z-q)}{(y-q)(z-p)}\] is a numerical constant, to be evaluated.
Prove that, if \(a\), \(b\), \(h\) are real numbers such that \(a > 0\), \(ab - h^2 > 0\), then \[ax^2 + 2hx + b > 0\] for all real values of \(x\). If \(p\), \(q\), \(r\) are real, investigate the conditions under which \[px^2 + 2qx + r > \rho\] for all real values of \(x\). Show that these conditions imply that \(r > 2|q|\).
Two points \(A\), \(B\) lie on a given circle; \(C\) is a point on one arc \(AB\) and \(D\) is a point on the other. The line through \(A\) parallel to \(CD\) cuts the line \(BD\) produced in \(\Gamma\) and the circle again in \(\Gamma'\). Prove that \(CD\Gamma\Gamma'\) is a parallelogram if, and only if, \(C\) is the middle point of the arc \(AB\). In the above configuration, the points \(A\), \(B\) (and consequently \(C\), the middle point of the arc \(AB\)) are regarded as given. Identify (i) the position of \(D\) if \(\Gamma'\) is the middle point of \(A\Gamma\), (ii) the centre of the circle on which \(\Gamma\) lies for varying positions of \(D\).
\(U\), \(V\), \(P\), \(Q\) are four points in order on a straight line, and circles are drawn on \(U\Gamma'\) and \(PQ\) as diameters. A direct common tangent touches the circle \(PQ\) at \(A\) and the circle \(UV\) at \(B\). Prove that the lines \(AP\), \(AQ\), \(BU\), \(BV\) lie along the sides of a rectangle \(AXBY\) whose centre is on the radical axis of the two circles and whose circumcircle passes through the limiting points \(L\), \(M\) of the coaxal system determined by them. Prove that the four points of intersection \((LX, MY)\), \((LY, MX)\), \((LA, MB)\), \((LB, MA)\) are at the vertices of a rectangle whose sides are parallel to those of \(AXBY\).
The edges \(a\), \(b\), \(c\), \(d\), \(p\), \(q\), \(r\), \(s\), \(t\), \(y\), \(z\), \(l\) of a cube are named as in the diagram, and \(f\), \(g\), \(r\), \(s\) are 'horizontal'; \(x\), \(y\), \(z\), \(l\) are 'vertical'. The cube is cut by a plane, the sections a, \(b\), \(c\), \(d\) (produced where necessary) meeting the plane at \(A\), \(B\), \(C\), \(D\), \(Z\), \(T\). Draw a clear annotated diagram of the section, showing the twelve points of intersection, and indicate which sets of more than two of them lie, and indicate which of those lines are parallel.
The circle whose centre is the point \(P(ap^2, 2ap)\) of the parabola \(y^2 = 4ax\) and which touches the \(x\)-axis meets the \(y\)-axis in points \(M\), \(N\). Prove that, for \(M\), \(N\) to be real and distinct, \(|p| < 2\). The tangents to the circle at \(M\), \(N\) meet in \(U\). Prove that \(PU\) is constant for all positions of \(P\), and that as \(P\) varies the polar of \(U\) with respect to the parabola touches a congruent parabola.
A rectangular hyperbola having the coordinate axes as asymptotes touches the ellipse \(b^2x^2 + a^2y^2 = a^2b^2\) at a point \(P\) in the first quadrant and at \(Q\) in the third. The foci of the ellipse are \(S_1\), \(S_2\) and the foci of the hyperbola are \(T_1\), \(T_2\). Prove that, if \(S_1S_2 = T_1T_2\) then the eccentricity \(e\) of the ellipse satisfies the equation \[e^4 + 4e^2 - 4 = 0,\] and verify that this equation has one and only one relevant root.
The hyperbola \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a, b > 0)\] has foci \(S(ae, 0)\), \(S'(-ae, 0)\); the foot of the perpendicular from \(S\) to the asymptote \[\frac{x}{a} - \frac{y}{b} = 0\] is \(P\). Find the lengths of the sides of the triangle \(OSP\), where \(O\) is the centre. The line \(PS\) meets the other asymptote in \(Q\), and \(U\) is the middle point of \(PQ\). Prove that the acute angle between the line \(OU\) and the \(x\)-axis is \(\alpha\), where \[\tan \alpha = b^3/a^3.\]
Two triangles \(ABC\), \(A'B'C'\) in general position in a plane are so related that \(AA'\), \(BB'\), \(CC'\) are in perspective from a point \(O\). The sides \(BC\), \(B'C'\) meet in \(L\); \(CA\), \(C'A'\) meet in \(M\); \(AB\), \(A'B'\) meet in \(N\). Prove that \(L\), \(M\), \(N\) are collinear. The line \(LMN\) meets \(AA'\) in \(U\), \(BB'\) in \(V\), \(CC'\) in \(W\). Prove that the pairs \(L\), \(U\); \(M\), \(V\); \(N\), \(W\) are in involution. In a particular case, \(L\), \(U\) coincide and \(M\), \(V\) coincide. Examine whether your proofs of the preceding results remain valid. If you decide that they are not, point out the deficiency, but you are not asked to formulate a fresh proof.
Show how to obtain the equation of a conic through the vertices \(X\), \(Y\), \(Z\) of the triangle of reference for general homogeneous coordinates in the form \[yz + zx + xy = 0.\] The tangent at \(X\) is met by \(YZ\) in \(P\); by the tangent at \(Y\) in \(V\); and by the tangent at \(Z\) in \(W\). The tangents at \(Y\), \(Z\) meet in \(R\), and \(YW\) meets \(ZV\) in \(Q\). Prove that the conic through \(Y\), \(Z\), \(Q\), \(R\) which touches \(XY\) at \(Y\) and \(XZ\) at \(Z\) touches at \(Q\) and \(R\) the conic through \(V\), \(W\), \(Q\), \(R\) which touches \(PQ\) at \(Q\) and \(PR\) at \(R\).