A circle cuts the conic \(Ax^2 + By^2 = 1\) in four points \(P_1\), \(P_2\), \(P_3\), \(P_4\). Establish a result about the directions of the lines \(P_1P_2\), \(P_3P_4\). If the conic is an ellipse and the eccentric angle of \(P_k\) is \(\alpha_k\) (\(k = 1, 2, 3, 4\)), prove that \(\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4\) is an integral multiple of \(2\pi\). Investigate the analogous result if the conic is a hyperbola and the coordinates of \(P_k\) are $$x = a\cosh u_k, \quad y = b\sinh u_k.$$
A point moves in space so that its distance from each of two intersecting straight lines is a given length \(l\). Prove that it lies on one of two ellipses which have a common minor axis.
Prove that the locus of the point $$\frac{x}{a_1t^2 + 2b_1t + c_1} = \frac{y}{a_2t^2 + 2b_2t + c_2} = \frac{1}{a_3t^2 + 2b_3t + c_3},$$ where the coefficients \(a_1\), \(\ldots\), \(c_3\) are real and \(t\) is a parameter, is, in general, a conic.
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.