\(P\) is any point on an ellipse whose centre is \(C\) and major axis \(AA'\). The angles \(PAQ, PA'Q\) are right angles. Prove that the locus of \(Q\) is an ellipse, and that if the normals at \(P\) and \(Q\) to the respective ellipses on which they lie meet \(AA'\) in \(G\) and \(H\), then \[ \frac{CG}{CH} = \frac{CB^2}{CA^2}, \] where \(B\) is one end of the minor axis of the original ellipse.
The lines \(CP\) and \(CQ\) are tangents to a conic at \(P\) and \(Q\); \(D\) and \(E\) are two other points on the conic. The line \(CD\) cuts \(PQ\) in \(G\), \(PE\) in \(H\) and \(QE\) in \(K\). By consideration of cross-ratios or otherwise prove that \[ \frac{CH.CK}{GH.GK} = \frac{CD^2}{GD^2}. \]
Prove that the asymptotes of a rectangular hyperbola bisect the angles between any pair of conjugate diameters. The centre of a rectangular hyperbola is also a focus of another conic \(T\). A pair of conjugate diameters of the hyperbola meet \(T\) in \(PQ, RS\). Prove that the poles with regard to \(T\) of \(PR, QR, PS, QS\) all lie on the asymptotes of the hyperbola.
Find the angle between the two straight lines \[ ax^2+2hxy+by^2=0. \] The distance between the feet of the perpendiculars from a point \(P\) on these lines is constant and equal to \(2c\). Prove that the locus of \(P\) is \[ (x^2+y^2)(h^2-ab) = c^2\{(a-b)^2+4h^2\}. \]
If the middle point of a chord of the parabola \(y^2=4ax\) lies on the line \(y=mx+c\), prove that the chord touches the parabola \[ (my+2a)^2=8ma(mx+c). \]
Find the equation of the normal to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at the point whose eccentric angle is \(\phi\). If the eccentricity of the ellipse is greater than \(\sqrt{2\sqrt{2}-2}\), prove that there are eight normal chords each of which meets the ellipse at extremities of a pair of conjugate diameters.
Find the condition that the pole of the line \(lx+my=1\) with respect to the conic \(ax^2+by^2=1\) should lie on the line \(px+qy=1\). A circle described on a chord \(PQ\) of this conic as diameter meets the conic again in \(R\) and \(S\). If the pole of \(PQ\) with respect to the conic lies on \(RS\), prove that \(PQ\) touches the conic \[ ax^2-by^2 = \frac{b-a}{b+a}. \]
Two straight lines cut the sides of a triangle \(ABC\) in \(P_1, Q_1, R_1\); \(P_2, Q_2, R_2\) respectively. Conics are circumscribed round the quadrilaterals \(Q_1R_1R_2Q_2, R_1P_1P_2R_2, P_1Q_1Q_2P_2\). Prove that three common chords of these conics are concurrent, and that each of them passes through an angular point of the triangle \(ABC\).
Find the sum of the cubes of the first \(n\) integers, and show that if \(m\) is the arithmetic mean of any \(n\) consecutive integers, the sum of their cubes is \[ mn\{m^2+\tfrac{1}{4}(n^2-1)\}. \] Prove that if \(s_1, s_2, s_3\) are the sums of the first, second and third powers of any consecutive integers \(9s_2^2 > 8s_1s_3\).
Denoting the number of combinations of \(n\) letters taken \(r\) together, all the letters being unlike, by \({}_nC_r\), show that the number of combinations taken \(n\) together, which can be formed from \(3n\) letters of which \(n\) are \(a\), \(n\) are \(b\), and the rest unlike is \[ {}_nC_n + 2{}_nC_{n-1} + 3{}_nC_{n-2} + \dots + n{}_nC_1 + (n+1); \] and show that this sum is \(2^{n-1}(n+2)\).