\(ABC\) is a triangle; \(D, E, F\) are the feet of the perpendiculars from \(A, B, C\) on the opposite sides. Prove that, when \(ABC\) is acute-angled, the perimeter of the triangle \(DEF\) is \(4R \sin A \sin B \sin C\), where \(R\) is the radius of the circle circumscribing \(ABC\). Examine the case of \(ABC\) obtuse-angled.
Find the necessary and sufficient condition that \[ ax^2+by^2+c+2fy+2gx+2hxy=0 \] may represent two straight lines. Prove that generally one conic and only one has the coordinate axes for its principal axes and touches the two straight lines. Shew that the points of contact lie on the line \[ x/f + y/g + 1/h = 0, \] and find the equation of the conic.
A uniform plank is to be lowered to the ground from a vertical position by one man, who places the lower end against a smooth vertical step and then walks backwards, exerting a force on the plank perpendicular to the length of the latter at a point which is always 6 feet above the ground. Show that the plank will slip if its length is greater than \(18\sqrt{3}\) feet.
Prove that in a position of equilibrium of a body under given forces, the potential energy is stationary in value; and obtain the criteria for stable and unstable positions of equilibrium. A picture hangs in a vertical plane from a smooth nail by a cord of length \(2a\), fastened to two symmetrically placed rings at a distance \(2c=2\sqrt{a^2-b^2}\) apart. Shew that, if the depth of the centre of gravity below the line of rings is greater than \(c^2/b\), the symmetrical position of equilibrium is the only one, and it is stable. Prove also that if the depth is \(c^2/b\), the symmetrical position is unstable.
Find graphically the positive root of \[ x = 2 \sin x \] in which the angle \(x\) is measured in radians. Prove that the number of real roots of the equation \(x = 10 \sin x\) is 7.
Prove that whatever value is given to \(\sigma\) the point \(Q\) at which \[ x = (a+b\sigma)\cos\phi, \quad y=(b+a\sigma)\sin\phi, \] lies on the normal at the point \(P\) of the ellipse \(x^2/a^2+y^2/b^2=1\) whose eccentric angle is \(\phi\). A circle with centre \(Q\) touches the ellipse at \(P\). Prove that the other two common tangents of the circle and ellipse meet at the point \begin{align*} (1-\sigma^2)x/\cos\phi &= a(1+\sigma^2)+2b\sigma, \\ (1-\sigma^2)y/\sin\phi &= b(1+\sigma^2)+2a\sigma. \end{align*}
A car is travelling at its maximum speed of 40 miles per hour on the level, the resistance being 160 lbs. weight per ton assumed to be independent of the speed. It then climbs a hill, gradient 1 in 25, and the speed falls until it is steady, the engine then working at the same effective H.P. as before. Find its steady speed up the hill. If the tractive force increases uniformly with the distance travelled up the hill, find the distance travelled before the uniform speed is attained.
Discuss the theory of frameworks consisting of light bars, smoothly pin-jointed, acted on by forces applied at the joints, treating the question of the determination of the stresses by graphical methods, by the method of virtual work, and by the method of sections. Discuss also (1) the conditions for determinateness of stress, and (2) the effect of friction at the joints. A triangular frame \(ABC\) is acted on by forces \(P, Q, R\) at the joints \(A, B, C\). Draw the diagram giving the stresses in the members. If the radius of each pin is given and the pins are supposed to fit loosely, give a diagram shewing the limits of variation of the stresses in the members assuming a coefficient of friction \(\mu\) between each pin and each rod.
Find the coordinates of the centres of circles which pass through the point \((1, 1)\) and touch the axis of \(x\) and the straight line \(3x + 4y = 5\).
Prove the anharmonic property of four fixed tangents of a conic, giving a few applications. Given a conic, centre \(O\), and three fixed points \(A, B, C\) upon it, determine (preferably by a geometrical construction) the point of the conic whose tangent meets those at \(A, B, C\) in three points \(a, b, c\) such that \(ab=bc\).