If \(AP\) and \(PB\) are two lines which represent the momenta of two smooth spheres before impact, shew that for some position of \(P'\), \(AP'\) and \(P'B\) will represent their momenta after impact, and shew how to find \(P'\) when the masses of the spheres, the direction of their common normal at impact and the coefficient of restitution are known. Two equal smooth spheres \(A\) and \(C\) with centres \(O\) and \(O'\) are in contact on a smooth table. A third equal sphere, \(B\), moves along a line perpendicular to \(OO'\) so that it impinges first with \(A\) and indefinitely soon afterwards with \(C\). Shew that if the spheres are inelastic and if a line \(PS\) represent the velocity of \(B\) before impact, then \(PQ, QR, RS\) will represent the velocities of the spheres \(A, B, C\) after impact, where \[ \angle QPS = \angle RSP = \frac{\pi}{6} \] and \(PQ\) is equal to half the projection of \(PS\) on \(PQ\) and \(SR\) is half the projection of \(SQ\) on \(SR\).
A bead moves on a smooth wire in the form of a parabola with its axis vertical and vertex upwards. Shew that the pressure on the wire is inversely proportional to the length of the normal intercepted between the parabola and its directrix.
Prove that, if the straight lines joining a point \(P\) to the vertices of a triangle \(ABC\) meet the opposite sides in \(D, E, F\), then \[ BD \cdot CE \cdot AF = DC \cdot EA \cdot FB. \] Prove also that, if \(P\) is the centre of the inscribed circle of the triangle, \[ BD \cdot CE + CE \cdot AF + AF \cdot BD = DC \cdot EA + EA \cdot FB + FB \cdot DC. \]
Prove that, if two tangents are drawn to an ellipse from an external point, they subtend equal angles at either focus. From the foci \(S, S'\) parallel lines \(SP, S'P'\) are drawn to the ellipse; the tangents at \(P\) and \(P'\) intersect in \(T\), and \(SP'\) and \(S'P\) intersect in \(Q\): prove that \(T\) is on the auxiliary circle, that \(TQ\) bisects the angle \(PQP'\), and that the projection of \(TQ\) on \(QP\) or \(QP'\) is equal to a fourth of the latus rectum.
Shew that the anharmonic ratio of a pencil from any point of a conic to four fixed points on the conic is constant. The tangents at \(P, P'\) to an ellipse cut the major axis in \(T, T'\) and the chord \(PP'\) cuts it in \(N\); prove that \[ CN (CT \cdot NT' + CT' \cdot NT) = CA^2 (NT + NT'). \]
Prove that the polar reciprocal of a circle with respect to another circle is a conic section. \(A\) and \(B\) are two circles; \(P\) is the polar reciprocal of \(A\) with respect to \(B\), and \(Q\) the polar reciprocal of \(B\) with respect to \(A\): shew that the ratio of the latera recta of \(P\) and \(Q\) is equal to the cube of the ratio of their eccentricities.
Prove that the planes which bisect at right angles the six edges of a tetrahedron pass through a common point.
From a point \((x',y')\) perpendiculars are drawn on the lines given by \[ ax^2+2hxy+by^2=0, \] the axes being at right angles; prove that the length of the perpendicular from \((x',y')\) on the line joining the feet of these perpendiculars is \[ \frac{ax'^2+2hx'y'+by'^2}{\{(a-b)^2+4h^2\}^{\frac{1}{2}}(x'^2+y'^2)^{\frac{1}{2}}}. \]
Prove that, if the normals at the points in which the conic \(ax^2+by^2=1\) is cut by the lines \(lx+my=1\) and \(l'x+m'y=1\) meet in a point, then \[ ll'/a=mm'/b=-1. \] Shew that, if one of these lines passes through a fixed point, the other touches a fixed parabola.
\(S=0\) is the equation of a conic, \(T=0\) the equation of a tangent, \(u=0\) the equation of a chord: interpret the equations \[ S-\lambda u^2=0, \quad S-\lambda uT=0, \quad S-\lambda T^2=0. \] Find the equation of the rectangular hyperbola which has contact of the highest order possible at a point of the ellipse \(x^2/a^2+y^2/b^2=1\); and shew that the locus of the centres of such rectangular hyperbolas is \[ \frac{(x^2+y^2)^2}{a^2+b^2} = \frac{x^2}{a^2}+\frac{y^2}{b^2}. \]