Problems

Find the locus of a point from which perpendicular tangents can be drawn to a given conic, noting any special cases. Hence by projection, or otherwise, shew that the locus of a point from which the tangents drawn to a given conic are harmonically conjugate with respect to the joins of the point to two fixed points is another conic.

Points are taken on a given line and through each the perpendicular to the polar of the point with respect to a given conic is drawn. Shew that these perpendiculars all touch a parabola which also has the given line as a tangent.

Prove that a pair of straight lines equally inclined to the axes of a central conic cuts it in four concyclic points, and that every pair of straight lines through these points is equally inclined to the axes. Hence deduce a construction for the centre of curvature at any point of a given central conic.

Shew that in general four fixed planes having a straight line in common intersect any straight line in four points having a fixed cross-ratio. \(l_1, l_2, l_3\) are three given straight lines. Shew that the cross-ratios of the ranges on these lines made by four common transversals are equal.

\(P\) is any point within a triangle \(ABC\), and at a distance \(d\) from its circumcentre, the circumradius being \(R\). If \(L, M, N\) be the feet of the perpendiculars from \(P\) drawn to the three sides \(BC, CA, AB\), shew that:

1. \(PL = (R-d)\cos A + \frac{d}{R}P'L'\), where \(P'\) is a certain point on the circumcircle and \(L'\) the foot of its perpendicular on to \(BC\).
2. The area of the triangle \(LMN\) is \(\frac{1}{4}(R^2-d^2)\sin A \sin B \sin C\).

Shew that if \(P_1, P_2, \dots, P_{2n}\) be vertices of a regular polygon with an even number of sides \(2n\) and \(O\) be the centre of the circumscribing circle, the product of the perpendicular distances from \(O\) on to \(P_1P_2, P_1P_3, \dots, P_1P_n\) is \((\frac{1}{2}a)^{n-1}\sqrt{n}\), where \(a\) is the radius of the circumscribing circle.

Find the locus of the centres of circles passing through a given point and cutting a given circle orthogonally.

\(A\) and \(B\) are two fixed points and \(\lambda\) is a fixed line through \(A\); a variable circle through \(A\) and \(B\) cuts \(\lambda\) again in \(P\). Prove that the tangent at \(P\) to this circle touches a fixed parabola with its focus at \(B\).

\(ABC\) is a triangle inscribed in a conic and the points \(Q\) and \(R\) on \(CA\) and \(AB\) respectively are conjugate with respect to the conic; prove that the lines \(QR\) and \(BC\) are conjugate with respect to the conic.

\(A\) and \(B\) are two fixed points and \(\lambda\) and \(\mu\) are two fixed lines in a plane; prove that the locus of a point \(P\), such that \(PA, PB\) are harmonically separated by the lines through \(P\) parallel to \(\lambda\) and \(\mu\), is a hyperbola, whose asymptotes are the lines through the middle point of \(AB\) parallel to \(\lambda\) and \(\mu\).