\(D, E, F\) are the middle points of the sides \(BC, CA, AB\) of a triangle \(ABC\), and points \(P, Q, R\) are taken in these sides. Prove that a necessary and sufficient condition that perpendiculars to the respective sides through \(P, Q, R\) should be concurrent is \[ BC \cdot DP + CA \cdot EQ + AB \cdot FR = 0. \] The perpendiculars from the vertices \(A, B, C\) of one triangle on the sides \(B'C', C'A', A'B'\) of a coplanar triangle are concurrent. Prove that the perpendiculars from \(A', B', C'\) on the sides \(BC, CA, AB\) respectively are also concurrent.
Prove that, if a pencil of four straight lines \(OA, OB, OC, OD\) is cut by a variable straight line in \(P, Q, R, S\) respectively, the cross-ratio of \(PQRS\) is constant. Prove that, if three points are given on a straight line, three other points can be determined each of which forms with the given three a harmonic range, and that the relation between the two sets of three points is a mutual one.
Prove that the polar reciprocal of a conic with respect to any circle whose centre is at a focus is another circle, and find the points into which the directrix and the asymptotes are reciprocated. \(D, E, F\) are the mid-points of the sides of a triangle \(ABC\), and \(O\) is the orthocentre of the triangle \(DEF\). Prove that the two conics having \(O\) as focus and inscribed respectively in the two triangles have equal eccentricities.
Prove that the locus of a point in space which is at the same given distance from each of two intersecting straight lines consists of two ellipses with a common minor axis.
Prove that, if a circle is described to touch the latus rectum of a parabola at the focus, four of the chords common to the circle and the parabola touch the circle which has the latus rectum for diameter.
Prove that the locus of a point, which is such that its polars with respect to two conics \(S, S'\) are perpendicular, is a conic. Prove that the locus will be a circle when, and only when, the asymptotes of \(S\) and \(S'\) taken together form two pairs of perpendicular lines.
Find the lengths of the axes of the conic \[ ax^2 + 2hxy + by^2 = 1. \] Prove that the ellipses, which pass through a fixed point and have a given director circle, have as their envelope an ellipse of which the given circle is the auxiliary circle and the fixed point is a focus. State the corresponding theorem for a parabola.
Obtain the equation of the circumcircle of the triangle of reference in areal coordinates \((x, y, z)\). Prove that the equation \[ p(q+r)yz + q(r+p)zx + r(p+q)xy = 0 \] will represent a rectangular hyperbola when the point \((p, q, r)\) is on the circumcircle of the triangle of reference.
Prove that, if \(r\) is prime to \(n\) and \(\alpha = \cos\frac{2r\pi}{n} + i \sin\frac{2r\pi}{n}\), the \(n\)th roots of unity are \(1, \alpha, \alpha^2, \dots, \alpha^{n-1}\). Shew that, if \(p\) is prime to \(n\), \[ 1 + \alpha^p + \alpha^{2p} + \dots + \alpha^{(n-1)p} = 0. \]
Differentiate \(\log\{x+\sqrt{(1+x^2)}\}\) with respect to \(x\). If \[ y = \frac{\log\{x+\sqrt{(1+x^2)}\}}{\sqrt{(1+x^2)}}, \] verify that \[ (1+x^2)\frac{dy}{dx} + xy = 1, \] and, assuming that \(y\) can be expanded in a series of ascending powers of \(x\), shew that the series is \[ x - \frac{2}{3}x^3 + \frac{2.4}{3.5}x^5 - \dots + (-1)^n \frac{2.4\dots2n}{3.5\dots(2n+1)}x^{2n+1} + \dots . \]