Given a circle of centre \(A\) and a point \(O\) outside it, obtain a construction, by ungraduated ruler and compasses, for a straight line \(OPQ\) cutting the circle in points \(P, Q\) such that \(P\) is the middle point of \(OQ\). What restriction is there on the position of \(O\) relative to the circle in order that the construction may be possible? If the points \(O, A\) are kept fixed but the radius of the circle allowed to vary, find the locus of \(P\).
The altitudes \(AD, BE, CF\) of an acute-angled triangle \(ABC\) meet in the orthocentre \(H\), and \(O\) is any point inside the triangle. The feet of the perpendiculars from \(O\) to \(AD, BE, CF\) are \(P,Q,R\) respectively. Prove that the triangle \(PQR\) is similar to the triangle \(ABC\).
\(A,B,C,D\) are four points in a plane. Prove that a necessary and sufficient condition for the pairs of lines \((AD,BC), (BD,CA), (CD,AB)\) to be perpendicular is that the middle points of the lines \(AD, BD, CD, BC, CA, AB\) should be concyclic. State and prove a corresponding result if \(A,B,C,D\) are not coplanar.
The foci of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] are \(S(ae,0), S'(-ae,0)\). Prove that the circle of centre \(S\) and radius \(2l\), where \(2l\) is the latus rectum of the ellipse \([l=a(1-e^2)]\), cuts the ellipse in two (real) points \(P,P'\), provided that \(e > \frac{1}{2}\). The line \(S'P\) is produced to \(Q\) so that \(PQ=PS\). Prove that \(SP\) bisects the angle \(S'SQ\).
Sketch the curve given parametrically by the equations \[ x=at^3, \quad y=3at. \] The chord joining the points \(P(ap^3, 3ap), Q(aq^3, 3aq)\) has constant gradient \(m\). Prove that the middle point of \(PQ\) lies on the curve \[ 8my^3 + 27a^2(mx-3y) = 0. \] Verify that this curve meets the given curve at the origin and at the two points where the tangent to the given curve is in the direction defined by \(m\).
The tangent to a hyperbola at a point \(P\) meets the asymptotes at \(L,M\). Prove that \(P\) is the middle point of \(LM\). Prove that, if the hyperbola is rectangular, the circle \(OLM\) (where \(O\) is the centre of the hyperbola) meets the axes of the hyperbola in points \(U,V\) such that \(L,M,U,V\) are the vertices of a square.
A straight line (not one of the axes of coordinates) touches the circle \[ x^2+y^2-2ax-2ay+a^2=0 \] and meets the axes of coordinates in \(P,Q\). Prove that the middle point of \(PQ\) lies on the rectangular hyperbola \[ 2xy - 2a(x+y)+a^2 = 0. \] Draw a sketch giving the relative positions of the axes of coordinates, the circle (taking \(a\) to be positive), the hyperbola and its axes.
An ellipse of semi-axes \(a\) and \(b\) (\(a>b\)) touches each of two fixed perpendicular lines in its plane. Prove that its centre must lie on one or other of four circular arcs, each of length \[ \sqrt{(a^2+b^2)} \{\tfrac{1}{2}\pi - 2 \tan^{-1}(b/a)\}. \]
\(ABC, A'B'C'\) are two triangles in perspective, so that \(AA', BB', CC'\) meet in a point \(O\). The corresponding 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. [For the purposes of this proof, the triangles may be taken to be coplanar or not, according to choice.] In a particular case, the point \(M\) lies on the line \(OBB'\) and the point \(N\) on the line \(OCC'\); the line \(LMN\) meets \(OAA'\) in \(U\). Prove that \(U\) is the harmonic conjugate of \(L\) with respect to \(M\) and \(N\). Interpret the last result when \(OAA'U\) is the "line at infinity".
Prove that the equation of the pair of tangents at the points of intersection of the conic \[ x^2+y^2+z^2=0 \] with the line \[ lx+my+nz=0 \] is \[ (m^2+n^2)x^2 + (n^2+l^2)y^2 + (l^2+m^2)z^2 - 2mnyz - 2nlzx - 2lmxy = 0. \] The condition for one of these tangents to pass through an arbitrary point \((\alpha,\beta,\gamma)\) is, after substituting \(\alpha,\beta,\gamma\) for \(x,y,z\) and rearranging, \[ (\beta^2+\gamma^2)l^2 + (\gamma^2+\alpha^2)m^2 + (\alpha^2+\beta^2)n^2 - 2\beta\gamma mn - 2\gamma\alpha nl - 2\alpha\beta lm = 0. \] Verify that this equation, when \(l,m,n\) are regarded as variable, is the tangential equation of a point-pair, and identify the two points.