A heavy particle of weight \(W\), attached to a fixed point by a light inextensible string, describes a circle in a vertical plane. The velocities at the highest and lowest points are \(V\) and \(nV\) respectively. Shew that, when \(\theta\) is the inclination of the string to the downward vertical, the tension in the string is \[ \left\{ \frac{2(n^2+1)}{n^2-1} + 3\cos\theta \right\} W. \] Hence find the limits within which \(n\) must lie if the string cannot support a tension greater than \(9W\).
A mass is attached to the lower end of a light elastic string \(AB\) of unstretched length \(a\), and an equal mass is attached to the first by a light inextensible string \(BC\). The upper end \(A\) of the string is fixed and the masses are pulled downwards until \(AB\) has increased its length to \(2a\), the tension being then three times the combined weight of the masses. The system is released and motion takes place in a vertical line. When the masses have reached the highest points of their motions, the lower one is removed by cutting the string \(BC\). Shew that the remaining mass comes to rest again at a height \(\frac{1}{2}(5-\sqrt{7})a\) above its initial point.
Prove the property of the ``Nine Point'' circle of a triangle. Shew that if through the mid-points of the sides of a triangle \(ABC\) straight lines are drawn perpendicular to the bisectors of the opposite angles, the triangle so formed has the same nine point circle as triangle \(ABC\).
Explain what is meant by a system of Coaxal Circles. Shew that any straight line is cut by the circles in a set of points in involution. Indicate the double points of the involution and shew that the circle with the line joining them as diameter is orthogonal to any circle of the system. Prove that if \(P, Q\) be two points on such an orthogonal circle which are also conjugate with respect to a given circle of the system of coaxal circles and do not both lie on the same diameter of it, then they are conjugate with respect to any circle of the system.
\(ABCD\) is a tetrahedron with a fixed base triangle \(ABC\) and a variable apex \(D\). Shew that the perpendiculars from \(A, B, C\) on to the opposite faces of the tetrahedron always intersect a fixed straight line perpendicular to the base and cutting it in the orthocentre of triangle \(ABC\). Prove that if \(D\) lies on this line the perpendicular from any apex on to the opposite face cuts it in its orthocentre.
Define the ``Cross Ratio'' \((ABCD)\) of four collinear points \(A, B, C, D\). Shew that the necessary and sufficient condition for the points to constitute a harmonic range is that \((ABCD)=(ADCB)\). Prove that the cross ratio of the pencil subtended by four fixed points on a conic at a variable point on it is constant in value, and that if this pencil be harmonic, the intersection of the tangents at two conjugate points of the four lies on the join of the other two.
Shew that the tangent to a conic is a bisector of the angle subtended by the two foci at the point of contact. Through a point \(P\) on an ellipse chords \(PQ, PR\) are drawn, one through each focus, to cut the curve again in \(Q, R\). The tangents at \(Q\) and \(R\) intersect in \(T\). Prove that \(PT\) is the normal to the curve at \(P\).
Find the equation of the normal to the parabola \(y^2 = 4ax\) at the point \((at^2, 2at)\). Three points \(P, Q, R\) are chosen on the parabola so that their normals intersect one another at a point on a fixed straight line perpendicular to the axis. Shew that the median point of triangle \(PQR\) is fixed. Find its coordinates in terms of the fixed line.
Determine the foci of the conic \(ax^2+2hxy+by^2+2gx+2fy+c=0\), and prove that the equation of the two straight lines through a focus which satisfy the condition of tangency has the form of the equation of a circle. Interpret this result in terms of the ``circular points'' of infinity, and show that a system of confocal conics may be regarded as inscribed in a certain quadrilateral.
Shew how by projection from a vertex, intersecting straight lines can be transformed into intersecting straight lines with the same angles between them as before. Prove that if two conics touch at \(A\) and \(B\), then the two polars of any point \(P\) with respect to the conics intersect on \(AB\).