A conic is drawn to touch four tangents to a given conic and the chord of contact of one pair of tangents; prove that it also touches the chord of contact of the other pair.
Prove that the common tangents to the two circles \begin{align*} x^2+y^2-2(a+b)x+c=0, \\ x^2+y^2-2(a-b)x+c=0, \end{align*} touch also the parabola \(y^2=4ax\). Express the equation to these common tangents in the form \[ (y^2-4ax)\{cy^2+(a^2+c)(c+a^2-b^2)\} + \{a(c+a^2)+x(c+a^2-b^2)\}^2 = 0. \]
Find the equation of a line perpendicular to the line \(lx+my+n=0\) and conjugate to it with respect to the ellipse \(x^2/a^2+y^2/b^2=1\), and show that the two lines determine, on the major axis of the ellipse, a pair of points harmonically related to the foci.
From any point on the normal to a rectangular hyperbola at a given point \(P\), the other three normals are drawn to the curve. Show that the locus of the centroid of their feet is the diameter of the hyperbola parallel to the normal at \(P\).
A given circle of radius \(r\) has its centre at the point \((c,o)\). A point \(P\) moves so that the length of the tangent from \(P\) to the circle bears a constant ratio \(e\) to the distance of \(P\) from the axis of \(y\). Prove that the locus of \(P\) is a conic of eccentricity \(e\). Show that the distances of the directrices of the conic from the line are the roots of the equation \[ e^2(1-e^2)x^2 - 2c e^2 x + r^2 = 0. \]
Show that if the sides of the pedal triangle of the triangle \(ABC\) be produced to meet the opposite sides in \(D, E, F\), the straight line \(DEF\) is the radical axis of the circumcircle of the triangle \(ABC\) and the circle with respect to which the triangle \(ABC\) is self-conjugate.
A regular hexagon \(ABCDEF\) of equal uniform rods each of weight \(W\) is suspended from \(A\). Equal weights \(W\) are hung from \(E\) and \(C\), and the frame is held rigid by light rods \(AE, AD, AC\). All rods are freely jointed at their ends. By drawing a force diagram, or otherwise, obtain the stresses in each of the rods \(AE, AD,\) and \(AC\).
A uniform rod \(AB\) of length \(2a\) can turn freely in a vertical plane about its midpoint \(O\). A weight \(W\) is attached at \(A\). An elastic string, of natural length \(2a\) and modulus \(\lambda\), has its ends attached to \(A\) and \(B\), and passes over a small smooth pulley at \(C\), a point vertically above \(O\) at a distance \(a\). Shew that the rod can be in equilibrium in an oblique position if \(W < \frac{\lambda}{2}\), and determine whether the equilibrium is stable.
If in the plane of a triangle \(ABC\), three forces act along and are proportional to \(AD, BE,\) and \(CF\) where \(D, E, F\) are points in \(BC, CA, AB\) respectively, shew that equilibrium is only possible if \(D, E, F\) are midpoints of the sides of the triangle. If \(n\) concurrent coplanar forces act along the lines \(OA_1, OA_2, \dots OA_n\), and are of magnitudes \(m_1.OA_1, m_2.OA_2, \dots m_n.OA_n\) respectively, prove that their resultant is represented in magnitude and direction by \(M.OG\) where \(G\) is the centre of mass of virtual masses \(m_1\) at \(A_1\), \(m_2\) at \(A_2, \dots m_n\) at \(A_n\), and \(M=m_1+m_2+\dots+m_n\). Extend this result to the case of parallel forces, and explain what happens in either case when \(M=0\).
A uniform chain is held against a smooth curve in a vertical plane. Shew that the difference in tension between the ends of the chain is the weight of a length of the chain which would hang vertically from the upper end down to the level of the lower. If the chain is in equilibrium in a vertical plane and rests on two smooth cylinders of unequal radii whose axes are perpendicular to the plane of the chain and do not lie in the same horizontal plane, prove that the ends of the chain both lie in the directrix of the catenary in which the part of the chain between the cylinders hangs.