The hyperbola \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a, b > 0)\] has foci \(S(ae, 0)\), \(S'(-ae, 0)\); the foot of the perpendicular from \(S\) to the asymptote \[\frac{x}{a} - \frac{y}{b} = 0\] is \(P\). Find the lengths of the sides of the triangle \(OSP\), where \(O\) is the centre. The line \(PS\) meets the other asymptote in \(Q\), and \(U\) is the middle point of \(PQ\). Prove that the acute angle between the line \(OU\) and the \(x\)-axis is \(\alpha\), where \[\tan \alpha = b^3/a^3.\]
Two triangles \(ABC\), \(A'B'C'\) in general position in a plane are so related that \(AA'\), \(BB'\), \(CC'\) are in perspective from a point \(O\). The 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. The line \(LMN\) meets \(AA'\) in \(U\), \(BB'\) in \(V\), \(CC'\) in \(W\). Prove that the pairs \(L\), \(U\); \(M\), \(V\); \(N\), \(W\) are in involution. In a particular case, \(L\), \(U\) coincide and \(M\), \(V\) coincide. Examine whether your proofs of the preceding results remain valid. If you decide that they are not, point out the deficiency, but you are not asked to formulate a fresh proof.
Show how to obtain the equation of a conic through the vertices \(X\), \(Y\), \(Z\) of the triangle of reference for general homogeneous coordinates in the form \[yz + zx + xy = 0.\] The tangent at \(X\) is met by \(YZ\) in \(P\); by the tangent at \(Y\) in \(V\); and by the tangent at \(Z\) in \(W\). The tangents at \(Y\), \(Z\) meet in \(R\), and \(YW\) meets \(ZV\) in \(Q\). Prove that the conic through \(Y\), \(Z\), \(Q\), \(R\) which touches \(XY\) at \(Y\) and \(XZ\) at \(Z\) touches at \(Q\) and \(R\) the conic through \(V\), \(W\), \(Q\), \(R\) which touches \(PQ\) at \(Q\) and \(PR\) at \(R\).
\(ABCD\) is a trapezium, with \(AB\) parallel to \(DC\). Lines \(BL\), \(DM\) are drawn to \(AC\), meeting the respective sides \(CD\), \(AB\) of the trapezium, produced if necessary, in \(L\) and \(M\). Prove that the centroid of the trapezium lies on \(LM\).
\(P\) and \(Q\) are two points on a semicircle whose diameter is \(AB\); \(AP\) and \(BQ\) meet in \(N\). Prove that the circle on \(MN\) as diameter cuts the semicircle orthogonally and that \(MN\) is perpendicular to \(AB\).
\(PP'\) is a focal chord of a parabola. Prove that the circle on \(PP'\) as diameter touches the directrix. If the normals to the parabola at \(P\), \(P'\) meet the curve again in \(Q\), \(Q'\) prove that \(PP'\) and \(QQ'\) are parallel.
Two parallel tangents of an ellipse, whose points of contact are \(P\) and \(P'\), are met by a third tangent in \(Q\) and \(Q'\). Prove that \(PQ \cdot P'Q'\) is equal to the square on the semidiameter conjugate to \(PP'\).
Prove that the locus of points from which the two tangents to the conic $$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$$ are perpendicular is the circle (the director circle of the conic) $$(ab - h^2)(x^2 + y^2) - 2(hf - bg)x - 2(gh - af)y + (bc + ca - f^2 - g^2) = 0.$$ If \(P\) is a point of a conic, \(Q\) the centre of curvature of the conic at \(P\), and \(R\) the image of \(Q\) in \(P\), prove that \(P\) and \(R\) are conjugate with respect to the director circle.
A triangle \(PQR\) is such that its vertices lie on the sides \(BC\), \(CA\), \(AB\), respectively, of a fixed triangle. Its sides \(PR\) and \(PQ\) pass through two fixed points \(M\), \(N\) on a fixed line through \(A\). Prove that \(QR\) passes through a fixed point \(L\), and identify this point precisely. State the dual theorem.
In a homography \(T\) on a straight line \(l\), to points \(A\), \(B\) there correspond respectively \(A'\), \(B'\), and \(M\) is a self-corresponding point. If \(M\) are any two points on a straight line through \(M\), \(BA\), \(B'A'\) meet in \(A'\) and \(BB\), \(B'B'\) meet in \(B'\). If \(A'B'\) meets \(l\) in a point \(N\) distinct from \(M\), prove that \(N\) is also a self-corresponding point of \(T\), and deduce that the cross-ratio \((M \text{ } NPP')\), where \(P'\) corresponds to \(P\), is constant for all positions of \(P\) on \(l\). What can be said about \(T\) if \(A'B'\) passes through \(M\)?