If \(A,B,C,D\) are four points on the same straight line, and circles are drawn through \(AB, BC, CD, DA\) to pass through a common point \(O\), prove that the product of the radii of the first and third circles is equal to the product of the radii of the second and fourth.
\(T\) is a point on a tangent at a point \(P\) of an ellipse so that a perpendicular from \(T\) on the focal distance \(SP\) is of constant length. Shew that the locus of \(T\) is a similar and similarly situated concentric ellipse.
If \[ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0 \quad \text{and} \quad x^2+y^2+z^2=0, \] prove that \[ \left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)(x+y+z)^3+9=0. \]
Find the value of \[ \begin{vmatrix} a^2-bc & b^2-\omega^2ca & c^2-\omega ab \\ c^2-\omega^2ab & \omega a^2-bc & b^2-ca \\ b^2-\omega ac & c^2-ab & \omega a^2-bc \end{vmatrix}, \] where \(\omega\) is an imaginary cube root of unity.
Eliminate \(\theta\) from the equations \[ b\cos(\alpha-3\theta)=2a\cos^3\theta, \quad b\sin(\alpha-3\theta)=2a\sin^3\theta. \]
The lengths of the perpendiculars from the angular points of a triangle on the straight line joining the orthocentre and the centre of the inscribed circle are \(p,q,r\). Prove that \(p\sin A, q\sin B, r\sin C\) are proportional to \[ \sec B-\sec C, \quad \sec C-\sec A, \quad \sec A-\sec B, \quad \text{respectively}. \]
Prove that the line joining the extremities of two variable radii of two given concentric circles which are equally inclined to a given direction is normal to a fixed conic.
Two equal smooth cylinders, each of radius \(a\), rest in parallel positions on a horizontal plane. On them rests an equilateral triangular prism in a symmetrical position touching the cylinders with two of its faces and with the third face horizontal. The cylinders are prevented from moving outwards by means of two stops in the plane. Prove that the height of each stop must be at least \[ a\left\{1-\frac{2\kappa+1}{2(\kappa^2+\kappa+1)^{\frac{1}{2}}}\right\}, \] where \(\kappa\) is the ratio of the weight of a cylinder to that of the prism.
A smooth circular cylinder of radius \(a\) is placed in a fixed position on a horizontal table. A heavy particle is placed at rest on the highest generator of the cylinder and is slightly displaced. Prove that the particle will strike the table at a distance \[ 5(\sqrt{5}+4\sqrt{2})a/27 \] from the line in which the cylinder touches the table.
A light elastic spring of natural length \(l\) and modulus \(\lambda\) is lying just stretched on a smooth horizontal table. One end is attached to a point of the table and the other is attached to a particle of mass \(m\). A blow of amount \(2\sqrt{m\lambda l/3}\) is applied to \(m\) perpendicularly to the spring. Find equations to determine the motion of \(m\), and shew that in the subsequent motion the greatest length of the spring is \(2l\).