Tangents \(PL, PM\) drawn to a parabola from a point \(P\) meet the directrix in \(U, V\) respectively. The second tangent from \(U\) meets \(PM\) in \(R\), and the second tangent from \(V\) meets \(PL\) in \(Q\). Prove that the point of intersection of \(QV, RU\) is the orthocentre of the triangle \(PQR\). Hence, or otherwise, show that the foot of the perpendicular from \(P\) to \(QR\) is the focus of the parabola.
The foci of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] are \(S(ae,0)\), \(S'(-ae,0)\), where \(e\) is the eccentricity, and \(P(x_1, y_1)\) is an arbitrary point of the ellipse. Prove that \[ SP = a-ex_1, \quad S'P = a+ex_1. \] Prove that, if \(p\) is the length of the perpendicular from the centre of the ellipse to the tangent at \(P\), then \[ p = \frac{ab}{\sqrt{(SP \cdot S'P)}}. \]
A straight line meets a hyperbola in \(A, B\) and its asymptotes in \(C, D\). Prove that the segments \(AB\) and \(CD\) have the same middle point. The tangent at a point \(P\) of a hyperbola, whose centre is \(O\), meets the asymptotes in \(U, V\). The normal at \(P\) meets the circle \(OUV\) in \(X, Y\). Prove that \(OX, OY\) are the axes of the hyperbola.
\(A, B, C, D\) are four points on a conic \(S\). The lines \(BC, AD\) meet in \(X\); the lines \(CA, BD\) meet in \(Y\); the lines \(AB, CD\) meet in \(Z\). Prove that the triangle \(XYZ\) (the diagonal triangle of the quadrangle \(ABCD\)) is self-polar with respect to \(S\). A general point \(Y'\) is taken on the line \(YZ\). The lines \(CY', BY'\) meet the conic \(S\) again in \(A', D'\), and the lines \(A'Y, D'Y\) meet \(S\) again in \(C', B'\). Prove that \(XYZ\) is also the diagonal triangle of the quadrangle \(A'B'C'D'\).
The tangents to a conic \(S\) at the points \(Z, X\) meet in \(Y\). Taking \(XYZ\) as triangle of reference, obtain the equation of the conic in the form \(y^2-zx=0\). The straight line \[ \lambda x + \mu y + \nu z = 0 \] meets the conic \(S\) in the points \(U, V\). Prove that the equation of the conic \(T\), through \(X, Y, Z, U, V\), is \[ \nu yz + \mu zx + \lambda xy = 0. \] Prove that, if the straight line \(UV\) passes through the fixed point \((a,b,c)\), then the pole of \(UV\) with respect to the conic \(T\) lies on the conic whose equation is \[ (cx+az)^2 = b(ayz+2bzx+cxy). \]
Perpendiculars \(PX, PY, PZ\) are drawn from an arbitrary point \(P\) in the plane to the sides of the triangle \(ABC\); the circle \(XYZ\) cuts the sides again in \(X', Y', Z'\). Prove that the perpendiculars to the sides at \(X', Y', Z'\) are concurrent at, say, \(P'\). Show that, if \(Q, R\) are two points such that the feet of the four perpendiculars from \(Q, R\) to the sides \(AB, AC\) are concyclic, then \(\angle QAC = \angle RAB\). Hence, or otherwise, prove that \(P, P'\) have the property that their joins to any vertex of the triangle \(ABC\) are equally inclined to the sides through that vertex.
A line in space cuts a plane at \(P\) and is perpendicular to two distinct lines lying in the plane and passing through \(P\). Prove that it is perpendicular to every line in the plane. The perpendicular from one vertex of a tetrahedron to the opposite face cuts it in the orthocentre of that face. Prove the same to be true for the other vertices.
From a variable point on a diagonal \(WY\) of a parallelogram \(WXYZ\) lines are drawn through fixed points \(B', C'\) on \(XY, YZ\) to cut the opposite sides in \(Q, R\). Prove that \(QR\) is parallel to \(B'C'\). The mid-points of the sides of the triangle \(ABC\) are \(A', B', C'\) and the altitudes are the lines \(p, q, r\); the circumcentre, orthocentre, centroid and nine-point-centre are \(O, H, G\) and \(N\). By applying the above proposition to the parallelogram formed by \(q, r\) and the perpendicular bisectors of \(AC, AB\), or otherwise, prove that, if \(E\) is a variable point on \(OH\) and if \(A'E, B'E, C'E\) cut \(p, q, r\) respectively in \(P, Q, R\), then the triangle \(PQR\) is similar to the triangle \(ABC\). Investigate the ratio \(QR:BC\) in the cases where \(E\) is at \(O, H, G, N\).
Define the polar of a point \((x_1, y_1)\) with respect to the circle \[ a(x^2+y^2)+2gx+2fy+c=0, \] and find its equation. Interpret the equation of the polar geometrically in the cases (i) \(a=0\), and (ii) \(ac=f^2+g^2\); verify algebraically both these interpretations.
Prove that in general the locus of points whose tangents to a given conic \(S\) are perpendicular is a circle \(\Sigma\). If \(S\) is neither a parabola nor a rectangular hyperbola, prove that the tangents to \(S\) from a point \(A\) on \(\Sigma\) cut \(\Sigma\) again at opposite ends of a diameter of \(S\) which is conjugate to the diameter through \(A\).