Find the equations of the tangent and normal at the point \((h,k)\) on the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. \] If the tangent at \(P\) cuts the directrices in \(Q\) and \(Q'\), prove that the other tangents from \(Q\) and \(Q'\) to the ellipse meet on the normal at \(P\).
Find the coordinates of the focus of the parabola \[ (x\sin\theta+y\cos\theta)^2=4ay\sin\theta, \] referred to rectangular axes; and shew that the equation of the latus rectum is \[ x\cos\theta-y\sin\theta=a\cos2\theta. \]
Find the conditions that the equation \[ Ax^2+By^2+Cz^2+2Fyz+2Gzx+2Hxy=0, \] in areal coordinates, represents a circle. Prove that, if \(ABC\) is the triangle of reference, \[ (y\cot B-z\cot C)^2=x(y+z) \] is the equation of the circle described on the perpendicular from \(A\) on the side \(BC\), as diameter; and find the coordinates of the point of intersection of the tangents to this circle at the points at which it cuts the sides \(AB\) and \(AC\).
Prove that, if \[ y=(\sin^{-1}x)^2, \] then \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = 2, \] and \[ (1-x^2)\frac{d^{n+2}y}{dx^{n+2}} - (2n+1)x\frac{d^{n+1}y}{dx^{n+1}} - n^2\frac{d^ny}{dx^n}=0. \]
The two chains of a suspension bridge hang in a parabola of span 80' and dip 16'; they are stiffened by a roadway, rigid except for hinges at the centre and at each end. If the chains and roadway weigh 0.3 ton per foot of span, find the maximum tension in each chain when a 2 ton concentrated load crosses the bridge, and the maximum bending moment on the roadway.
The countershaft of a lathe carries two gear-wheels whose pitch diameters are 6" and 3" respectively: the central planes of the wheels are 6" apart and the ends of the shafts run in bearings outside the wheels whose central planes are 2" from the central plane of the corresponding wheel. The gear-wheels on the mandrel have pitch diameters of 3" and 6", so that the total reduction ratio is 4:1, and the driving belt exerts a couple of magnitude 20 lb. ft. on the first of these two latter wheels. If the angle of obliquity of thrust between the gear teeth be 20\(^\circ\), calculate the load on each bearing of the countershaft, neglecting friction throughout.
State and prove any graphical construction for finding the acceleration of a steam-engine piston at any point of the stroke. Find the acceleration of the middle point of the connecting rod of an engine running at 400 r.p.m. at the instant when the crank is at right angles to the line of stroke, if the lengths of crank and connecting rod be 6" and 2' 0" respectively.
A circular galvanometer coil has a rectangular cross-section, the external and internal radii being 2 cm. and 1.5 cm. respectively, whilst the axial length is 1 cm. The coil is hung up by a bifilar suspension consisting of two light parallel strips each 15 cm. long, the strips being 0.5 cm. apart. Calculate the time of a small oscillation about a vertical basis midway between the strips.
A variable speed friction gear consists of a flat disc on a shaft running at uniform speed, in contact with the face of which runs a friction wheel on a shaft which intersects the first shaft at right angles, the wheel being pressed on to the disc with a pressure just sufficient to prevent slipping as a whole. Shew that the efficiency of the transmission is \(1/\left(1+\frac{a}{4r}\right)\), where \(a\) is the length of the line of contact of the friction wheel and \(r\) is the radius from the centre of the disc to the centre of that line of contact.
A uniform rod 8' long standing vertically on the ground falls over so that its centre strikes a horizontal bar: the bar is perpendicular to the length of the rod at the moment of impact and is 2' above the ground. Find the velocity with which the upper end of the rod strikes the ground, assuming that neither rod nor bar is bent by the impact.