Shew that the relation independent of \(\lambda\), which is satisfied by the roots of the quadratic \(az^2+bz+c+\lambda(a'z^2+b'z+c')=0\), is \[ \left(z_1-\frac{\beta}{\gamma}\right)\left(z_2-\frac{\beta}{\gamma}\right) = \frac{\beta^2-\alpha\gamma}{\gamma^2}, \] where \(\alpha=bc'-b'c, \beta=ca'-c'a, \gamma=ab'-ba'\). Deduce that, if \(a, b, c, a', b', c'\) be real numbers, the roots are real for all real values of \(\lambda\), provided \(\beta^2-\alpha\gamma\) be negative; but that, when \(\beta^2-\alpha\gamma\) is positive, there are real values of \(\lambda\) for which the roots are imaginary and that the points representing them in the Argand diagram lie on the circle \(\left(x-\frac{\beta}{\gamma}\right)^2+y^2 = \frac{\beta^2-\alpha\gamma}{\gamma^2}\).
Find the equation of the chord of the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] which has \((x',y')\) for its mid-point. Points \(P, Q\) are taken on a fixed conic \(C\) and points \(P', Q'\) on a second fixed conic \(C'\), so that \(PQ\) and \(P'Q'\) cut at right angles and bisect one another at \(R\). Shew that, in general, the locus of \(R\) is a conic; but, if the Cartesian equation to this locus is of the first degree, prove that \(C\) and \(C'\) are either parabolas whose axes are at right angles or rectangular hyperbolas such that each axis of one is parallel to an asymptote of the other.
Prove that the curve \(x=at^2-2bt+c, y=a't^2-2b't+c'\), where \(t\) is a variable parameter, is a parabola, and find the equation of the tangent at the point whose parameter is \(t\). Find the value of \(t\) at the vertex of the parabola, and prove that the values of \(t\) at the end of the latus rectum are \[ \frac{ab+a'b' \pm (ab'-a'b)}{a^2+a'^2}. \]
The normals at two points \(P, Q\) of a plane curve intersect in \(N\): shew that in general \((PN-QN)\) is of order \(s^2\), when the arc \(s\), between \(P\) and \(Q\), is regarded as small: but that the difference between \(PN\) and the radius of curvature at \(P\) is of order \(s\). The tangent at \(P\) to a plane curve meets a fixed straight line in \(T\); if the curve is such that the length \(PT\) is constant, prove that the centre of curvature at \(P\) lies on the perpendicular at \(T\) to the fixed straight line.
Explain the reduction of a system of coplanar forces to a single force or to a couple. If two forces \(P, Q\) act at fixed points \(A, B\) and have a resultant \(R\), show that if \(P\) and \(Q\) are turned through any the same angle, the resultant passes through a fixed point \(C\), such that the sides of the triangle \(ABC\) are proportional to \(P, Q, R\). Deduce the existence of such a point for \(n\) forces acting at given points. Examine the case where the \(n\) forces are in equilibrium.
A uniform bar \(PQ\) hangs from two fixed points \(A, B\) (by two equal crossed strings) with \(AB\) and \(PQ\) horizontal as in the sketch. With the axes as indicated shew that when \(PQ\) is displaced in the vertical plane of the figure so as to make an angle \(\theta\) with the axis of \(x\) (and \(\frac{1}{2}\pi-\theta\) with the axis of \(y\)), the coordinates of its centre are given by \[ x^2+y^2=h^2+4ab\sin^2\frac{1}{2}\theta, \] \[ x(a+b\cos\theta)+by\sin\theta = 0, \] where \(h\) is the depth of \(PQ\) below \(AB\) in equilibrium. Shew that the equilibrium is stable for small displacements in the plane of the figure if \(bh^2>a(a+b)^2\).
A particle is oscillating in a straight line, and its velocity \(v\) is connected with the displacement \(x\) (measured in a fixed direction from a fixed origin on the line) by the equation \[ v^2 = ax^2+2bx+c. \] Shew that, if \(a\) is negative and \(b^2>ac\), the motion is simple harmonic: and find the period and amplitude of the motion. A railway truck travelling with velocity \(V\) runs into a fixed stop, with a compressible buffer. Shew that if the buffer is compressed a distance \(l\) and the compression is always proportional to the pressure, the duration of contact is \(\pi l/V\).
Shew that the velocity of the bob of a simple pendulum at its lowest point, when making small vibrations, is approximately proportional to the arc of oscillation. Two balls are suspended by long strings from points at the same level so that when in equilibrium they are in contact with the line of centres horizontal. Explain how by experiments on the collision of the balls the idea of mass may be derived and the theory of momentum verified. Explain also how to determine by experiment the value of gravity and to verify that weight is proportional to mass.
Solve the equations: \[ x^2+y+z = y^2+z+x = z^2+x+y = 3. \] Eliminate \(x,y,z\) from the equations: \begin{align*} x+y+z &= a, \\ x^2+y^2+z^2 &= b, \\ x^3+y^3+z^3 &= c, \\ x^4+y^4+z^4 &= d. \end{align*}
Prove that, if \(a, b, c\) are different positive quantities, \[ a^3+b^3+c^3 > abc(a+b+c), \] and \[ \frac{a^3+b^3+c^3}{a^2+b^2+c^2} > \frac{a+b+c}{3}. \]