Problems

A line \(l\) is drawn through \(O\), the orthocentre of a triangle \(ABC\) and meets \(BC, CA, AB\) in \(D, E, F\) respectively. \(AO, BO, CO\) meet the circumcircle of \(ABC\) in \(P, Q, R\) respectively. Prove that \(DP, EQ, FR\) meet in a point \(S\) of the circumcircle, and that the parabola with \(S\) as focus and \(l\) as directrix touches \(BC, CA, AB\).

Prove that one conic confocal with \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) can be drawn to touch a general line. \par Tangents are drawn from a point \((h, k)\) to this system of confocal conics. Prove that the locus of points of contact is \[ \frac{x}{y-k} + \frac{y}{x-h} = \frac{a^2-b^2}{hy-kx}. \]

The polars of \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\) with respect to the conic \(ax^2+by^2=1\) meet this conic in \(Q_1R_1\) and \(Q_2R_2\) respectively. Shew that the six points \(P_1Q_1R_1P_2Q_2R_2\) lie on a conic, say \(K\). \par If \(P_1\) is kept fixed, and \(P_2\) describes a line, prove that the conics \(K\) all pass through a fixed point on this line.

Prove that on a straight line there is in general one pair of points conjugate with regard to all the conics through four points \(A, B, C, D\). Hence, or otherwise, shew that, if \(D\) is the orthocentre of the triangle \(ABC\), then all the conics are rectangular hyperbolas.

Obtain the equation of the circumcircle of the triangle formed by the three lines \[ ax+by+c=0, \quad a'x+b'y+c'=0, \quad a''x+b''y+c''=0. \] Find also the equation of the circle which has this triangle as a self-conjugate triangle.

Prove that in polar coordinates \((r, \theta)\) the radius of curvature of a curve is given by \[ \frac{\left\{r^2 + \left(\frac{dr}{d\theta}\right)^2\right\}^{3/2}}{r^2+2\left(\frac{dr}{d\theta}\right)^2 - r\frac{d^2r}{d\theta^2}}. \] Prove that the radius of curvature of the curve \(r=a(1-\cos\theta)\) is \(\frac{4a}{3}\sin\frac{\theta}{2}\). Sketch the curve.

Prove that in general a system of coplanar forces can be reduced to a force acting at an assigned point in the plane together with a couple \(G\). \par \(ABCD\) is a square of side \(a\). Forces \(1,2,3,4,P,kP\) act in \(AB, BC, CD, DA, AC, DB\) respectively. Shew that the locus of a point which moves so that \(G\) is constant is a straight line which passes through the same point \(H\) in \(BD\) whatever the value of \(k\). Determine the ratio in which \(BD\) is divided by \(H\) when \(G=\frac{1}{2}aP\sqrt{2}\) in the sense \(ABCD\).

A uniform ladder of weight \(w\) and length \(2l\) is placed with one end on the ground and the other end against a vertical wall. The ground slopes upwards towards the wall with which it makes an angle \(\frac{\pi}{2}+\alpha\) and the ground and wall are equally rough, \(\lambda(>\alpha)\) being the angle of friction. The ladder makes angles \(\theta\) and \(\frac{\pi}{2}-\alpha-\theta\) with the wall and ground respectively. \par Shew that the greatest distance up the ladder which a man of weight \(W\) may ascend without causing the ladder to slip is \[ \frac{l[2(w+W)\sin(\lambda-\alpha)\cos(\theta-\lambda)-w\sin\theta\cos\alpha]}{W\sin\theta\cos\alpha}. \]

Explain the use of Bow's notation in graphical statics. \par The diagram represents a pin-jointed framework of light rods held at two points \(A\) and \(O\) in the same vertical line. Determine by graphical methods the stress in each rod and the angle which the reaction at \(O\) makes with \(OA\).

An elastic string \(OA\), of mass \(m\) and coefficient of elasticity \(\lambda\), has when unstretched a length \(l\) and uniform line density. The string hangs in equilibrium from \(O\). Prove that the total extension of the string is \(\frac{mgl}{2\lambda}\). Shew further that the potential energy of the string is less by an amount \(\frac{mgl}{2}+\frac{m^2g^2l}{6\lambda}\) than when it is coiled up at \(O\).