Sum the series
Prove that the equation \[ Ax^2+Ay^2+2Gx+2Fy+C=0 \] represents a circle. Find the coordinates of its centre, its radius and the length of the tangent to the circle from the point \(h, k\). Find the locus of the centre of a circle which cuts \[ x^2+y^2+2Gx+2Fy+C=0 \quad \text{and} \quad x^2+y^2+2gx+2fy+c=0 \] orthogonally.
Shew that the coordinates of any point on a hyperbola can be expressed as \(a\sec\theta, b\tan\theta\); and further that the normal at this point has for its equation \[ ax\cos\theta+by\cot\theta=a^2+b^2. \] Hence prove that four normals can be drawn from any point to a hyperbola, and that the sum of the angles corresponding to the feet of these normals is an odd multiple of \(180^\circ\).
Prove that the equation \[ l=r(1+e\cos\theta) \] represents a conic whose focus is the pole. Find the equation of the tangent at the point whose vectorial angle is \(\alpha\). Shew that the pole of a chord, which subtends a constant angle at the focus, lies on a conic having the same focus and directrix.
Prove that the straight lines \[ ax^2+2hxy+by^2=0 \] are conjugate diameters of the conic \[ Ax^2+2Hxy+By^2=1 \] if \[ Ab+Ba=2Hh. \] Hence shew that the asymptotes are the double rays of the involution formed by pairs of conjugate diameters of a conic.
Prove that the moment of the resultant of a system of forces, acting in one plane on a rigid body, about a point in the plane, is equal to the algebraic sum of the moments of the forces. A circular tray of radius \(a\) stands on a single circular foot of radius \(b\). If \(w\) is the whole weight of the tray and its support, find how far from the centre a weight \(W\) can be placed without the tray falling over.
Prove that two couples, acting in one plane upon a rigid body, are in equilibrium if their moments are equal and in opposite senses. A uniform bar of weight \(W\) and length \(2a\) is suspended, from two points in a horizontal plane, by two equal strings of length \(l\), which are originally vertical: shew that the couple, which must be applied to the bar in a horizontal plane, to keep it at rest at right angles to its former direction is \[ \frac{Wa^2}{\sqrt{l^2-2a^2}}. \]
State the laws of friction and shew how they may be verified experimentally. A weight is pulled up a rough inclined plane by a rope parallel to a line of greatest slope; find the mechanical advantage and the efficiency of this machine.
Explain the terms Stable, Unstable and Neutral Equilibrium. A solid circular cylinder of radius \(a\) and height \(h\) has one end in the shape of a hemisphere; find the condition that it will be in stable equilibrium when standing on that end, on a smooth horizontal plane, with its axis vertical.
Reciprocate, with respect to the focus, the theorem that the circumcircle of the triangle formed by three tangents to a parabola passes through the focus. Generalise both theorems by projection.