If \(\alpha, \beta\) are two of the roots of the cubic equation \(x^3 + 3qx + r = 0\), prove that \(\alpha/\beta\) is one of the roots of the sextic equation \[ r^2(x^2+x+1)^3 + 27q^3x^2(x+1)^2 = 0. \] Explain this result in relation to the roots of the cubic equation, when (i) \(q=0\), (ii) \(r=0\), (iii) \(4q^3+r^2=0\).
Shew that by a suitable choice of the triangle of reference the homogeneous co-ordinates \((x, y, z)\) of any four points may be taken as \((f, \pm g, \pm h)\) and that the locus of the poles of the line \(lx+my+nz=0\) with respect to conics through these four points is given by \(f^2l yz + g^2m zx + h^2n xy = 0\). Explain this result when \(l=0\). Shew also that the locus of centres of conics through four fixed points is a parabola, if and only if, one of these points is at infinity.
The rectangular Cartesian coordinates \((x,y)\) of a point are given by \(x=at^2+2pt\), \(y=bt^2+2qt\), where \(a, b, p, q\) are constants and \(t\) is a parameter; prove that the locus of the point is a parabola, and that this parabola touches the line \(lx+my+1=0\), if \[ (pl+qm)^2 - (al+bm) = 0. \] Find (not necessarily in this order) the equations of the tangent at the vertex, of the axis and of the directrix, and the coordinates of the focus of this parabola. Verify from your results that the length of the latus rectum is \(4(aq-bp)^2/(a^2+b^2)^{3/2}\).
A cycloid may be defined as the locus of a point on the rim of a wheel of radius \(a\), which rolls without slipping along a horizontal straight line, the plane of the wheel being vertical; from this definition prove that the coordinates of any point on the cycloid may be written in either of the forms:
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On a rough inclined plane are placed a uniform block in the shape of a rectangular parallelepiped, of weight \(W\), with an edge horizontal, and, above the block and resting against it with the point of contact at the centre of a face of the block, a uniform sphere of weight \(W\) and radius \(a\). The edges of the block parallel to a line of greatest slope are of length \(3a\); the other edges are of length \(2a\). The coefficient of friction at all pairs of surfaces in contact is \(\frac{1}{2}\). The inclination \(\alpha\) of the plane to the horizontal is gradually increased from a very small value. Shew that when equilibrium is first broken the sphere does not begin to slip down the plane with its point of contact with the block unchanged, but rolls down the plane and slips against the block; and that the block slides and does not tilt. Shew also that equilibrium is first broken when \(\tan\alpha = \frac{1}{4}\).
A smooth wire in the form of a circle is placed in a vertical plane, and a bead of weight \(W\) which slides on it is attached to the highest point of the circle by a weightless elastic string of modulus \(kW\), whose natural length is equal to the radius of the circle. Find (i) the condition that the lowest position of the bead should be one of stable equilibrium, (ii) the condition that other equilibrium positions should be possible, (iii) where the other equilibrium positions are and whether or not they are stable.
A particle is projected along the outside surface of a smooth sphere of radius \(a\) ft. from the highest point with velocity \(\frac{1}{2}\sqrt{(ga)}\). Prove that it strikes a horizontal plane through the centre of the sphere at a distance \[ \frac{9\sqrt{39} + 7\sqrt{7}}{64} a \text{ ft.} \] from the centre.
A particle of mass \(m\) moves in a straight line on a rough horizontal table, under the influence of the frictional force, which has the constant magnitude \(n^2ml\), and a force of magnitude \(n^2mx\) towards a point O in the line of motion, where \(x\) is the displacement from O. The particle is released from rest at a distance \(15.5l\) from O. Shew that it performs simple harmonic half-oscillations about each of two points in turn, and that it comes finally to rest after making an integral number of half-oscillations; and find how many half-oscillations it makes before it comes finally to rest.
Two equal masses are fixed to a light rod, one at the top point and one at the middle point, and the lower end of the rod rests on a rough horizontal table. The rod is released from rest at an angle \(\alpha\) to the vertical. Shew that the lower end will not slip initially, for any value of \(\alpha\), if the coefficient of friction exceeds \(9/(2\sqrt{10})\).