A particle can move in a smooth circular tube which revolves about a fixed vertical tangent with uniform velocity \(\omega\). Find the position of relative equilibrium of the particle, and shew that the time of a small oscillation about that position is \[ \frac{2\pi}{\omega}\left\{\frac{\sin\alpha}{1+\sin^2\alpha}\right\}^{\frac{1}{2}}, \] where \(\alpha\) is the angle of inclination to the vertical of the radius to the particle when in relative equilibrium.
A point \(P\) is taken within a triangle \(ABC\), whose sides are \(a, b, c\), such that \(\frac{AP}{a} = \frac{BP}{b} = \frac{CP}{c} = \lambda\). Prove that \(A = 6\lambda^2\).
An ellipse has focus \(S\) and centre \(C\). The minor axis \(BB'\) meets a tangent in \(L\), and a parallel focal chord in \(K\). Prove that \[ SK=e.SL, \quad CK^2-c^2 = e^2.LB.LB'. \]
Prove that \[ \begin{vmatrix} \sin 2A & \sin C & \sin B \\ \sin C & \sin 2B & \sin A \\ \sin B & \sin A & \sin 2C \end{vmatrix} = 0, \] when \(A, B, C\) are angles of a triangle, and find the factors of \[ x^2\sin 2A + y^2\sin 2B + z^2\sin 2C + 2yz\sin A + 2zx\sin B + 2xy\sin C=0. \]
Prove that the number \((r+1)^2(r-1)\), when expressed in the scale of \(r\), is multiplied by \(r-1\) when its digits are reversed.
Prove that, when \((1+x)^5(1-x)^{-2}\) is expanded in powers of \(x\), the coefficient of \(x^{r+4}\) is \(16(2r+5)\), and that the sum of the first \(r+4\) coefficients is \(16(r^2+4r+5)\).
A chord of the circle \(x^2+y^2-my=0\) touches the ellipse \(x^2+\frac{1}{2}y^2-my=0\). Prove that its length is equal to its distance from the origin.
Find the equation of the pair of tangents drawn from the origin to the general conic. Shew that the tangents drawn from the fixed point \((-a, a)\) to the circle \[ x^2+y^2-2rx=0 \] coincide with those drawn to the circle \(x^2+y^2-2sy=0\) if \(r, s\) satisfy a single condition, and that the equation of the tangents is then \[ \left(\frac{x+a}{r}+\frac{y-a}{s}\right)^2 = \frac{(x+a)^2}{r^2} + \frac{(y-a)^2}{s^2}. \]
Find the conditions that the normals of an ellipse at the extremities of the chords \(lx+my+1=0, l'x+m'y+1=0\) should meet in a point. Express the coordinates of the point.
Prove that the locus of the centre of a conic which touches four given straight lines is itself a straight line.