Shew that a plane section of a circular cone satisfies the focus-directrix definition of a conic, and that the eccentricity is equal to \(\cos\theta \sec\alpha\), where \(\theta\) is the angle between the plane and the axis of the cone, while \(2\alpha\) is the vertical angle of the cone: shew further that the latus-rectum of the section is equal to \(2p \tan\alpha\), if \(p\) is the perpendicular from the vertex of the cone on the plane of the section. Prove that an ellipse or parabola of given size can be cut from a cone of any assigned angle; and discuss to what extent this is possible for a hyperbola.
Two planes (\(\alpha, \alpha'\)) cut at right angles in a line \(m\) and the point \(V\) is the vertex of a conical projection from \(\alpha\) to \(\alpha'\). A third plane \(\beta\) is drawn through \(V\) to cut both \(\alpha\) and \(\alpha'\) at right angles, in the lines \(l\) and \(l'\) respectively. Shew that the formulae of projection may be written in the forms \[ x' = \frac{f'x}{x-f}, \quad y'=\frac{-fy}{x-f}, \] where \(l\) is taken as axis of \(x\), \(l'\) as axis of \(x'\), and the axes of \(y, y'\) coincide along \(m\); \(f\) is the distance of \(V\) from the plane \(\alpha'\) and \(f'\) is its distance from \(\alpha\). Shew that, if \(c=f-f'\), any line through the point \(N(x=c, y=0)\) is projected into a line through \(N'(x'=-c, y'=0)\) making the same angle with the line \(l'\) as the original line does with \(l\); and that any circle with \(N\) as centre is projected into a conic with \(N'\) as focus and the line \(x'-f'=0\) as directrix.
Observations of a variable \(x\) are made at equidistant intervals of time; suppose that the values \(x_1, x_2, x_3\) correspond to the times \(t=-c, t=0, t=+c\), respectively, and find a quadratic expression in \(t\) to represent \(x\). A variable magnitude was observed as follows:
The diameter of a sphere is divided into two parts (of lengths \(p,q\)) by a perpendicular plane which divides the sphere into segments of volumes \(V, V'\) and of spherical surfaces \(S, S'\). Find (by the Integral Calculus or otherwise) formulae for \(V, V'\) and \(S, S'\) in terms of \(p, q\). Obtain the theorems of Archimedes
In the theory of ``meridional parts,'' the function \(y\) corresponding to a given latitude \(\theta\) is defined by the equation \[ \frac{dy}{dx} = \frac{1}{\cos\theta} \frac{d\theta}{d\phi}, \] where \(x\) is the number of minutes of arc in the longitude (whose circular measure is \(\phi\)). Shew that, if \(y\) is zero when \(\theta=0\), \[ y = k \log_e \left( \frac{1+\sin\theta}{\cos\theta} \right), \] where \(k\) is a constant to be found. From the four-figure tables prove that in latitude 30\(^\circ\), \(y=1888\). [Take \(\log_e 10=2.303, \log_{10}\pi = 0.4971\).]
Explain what is meant by the acceleration of a moving point (i) when it is moving in a straight line, (ii) when it is moving in a curve. If the velocity of a particle is made up of two components, \(u\) and \(v\), each fixed in direction but varying in magnitude, shew that the acceleration is made up of two components \(\frac{du}{dt}\) and \(\frac{dv}{dt}\) along the directions of \(u\) and \(v\) respectively. \(AB, BC\) are two rods of lengths \(a, b\) hinged together at \(B\), and to a fixed pivot at \(A\). \(AB\) is turning with uniform angular velocity \(\omega\) and \(BC\) with uniform angular velocity \(\omega'\). Shew that \(C\) will be at a point of inflexion on its path when \[ \cos \angle ABC = -\frac{a^2\omega^3+b^2\omega'^3}{ab\,\omega\omega'(\omega+\omega')}. \] (Note: Transcribed from image, differs from OCR.)
A convex quadrilateral is inscribed in a circle of given radius \(R\), and one side subtends a given angle \(\alpha\) at a point of the arc of the circle on the opposite side to the quadrilateral. Prove that the greatest area of the quadrilateral is \[ 2R^2 \sin^3\frac{2}{3}\alpha. \]
If \[ Al^2+Bm^2+Cn^2+2Fmn+2Gnl+2Hlm=0, \] find the coordinates of the point of contact of the line \(lx+my+n=0\) with its envelope. Shew that the equation \[ \frac{\lambda}{lx_1+my_1+n} + \frac{\mu}{lx_2+my_2+n} + \frac{\nu}{lx_3+my_3+n} = 0 \] is the tangential equation of a conic inscribed in the triangle whose vertices are \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), and that the points of contact divide the sides in the ratios \(\lambda:\mu:\nu\). If the conic is a parabola, prove that its axis is parallel to the line \[ \frac{x}{\lambda x_1+\mu x_2+\nu x_3} = \frac{y}{\lambda y_1+\mu y_2+\nu y_3}. \]
A uniform rough plank of weight \(W\) and thickness \(2b\) rests horizontally in equilibrium across a fixed rough cylinder of radius \(a\), and a particle of weight \(w\) is fixed to the plank vertically above the axis of the cylinder. Prove that, if \((W+w)a > b(W+2w)\), equilibrium is stable, and that if \((W+w)a < b(W+2w)\) it is unstable. Shew that in the former case there are two oblique positions of equilibrium, which are unstable, provided that the friction is great enough to prevent slipping.
A heavy uniform chain of length \(l\) hangs in equilibrium over the edge of a smooth horizontal table, and the end of the chain which is upon the table is attached to a spring of strength such that a force equal to the weight of the chain would stretch it through a length \(a\) (\(