Problems

Prove the formula \(\rho=r\frac{dr}{dp}\) for the radius of curvature of a curve given in terms of \(p\) the perpendicular on to the tangent from the origin, and \(r\) the radius vector. Obtain the \((p,r)\) equation of a parabola referred to its focus as origin in the form \(a.r=p^2\), and deduce that if the parabola is made to roll without slipping on a fixed straight line, its focus describes a curve whose radius of curvature is equal to the focal radius of the parabola at the corresponding instantaneous point of contact.

If \(X, Y, x, y\) are real quantities connected by the complex relation \[ Z = X+iY = f(x+iy) = f(z), \] where \(i=\sqrt{-1}\), shew that \[ J = \frac{\partial X}{\partial x}\frac{\partial Y}{\partial y} - \frac{\partial X}{\partial y}\frac{\partial Y}{\partial x} = (|f'(z)|)^2. \] Prove that if \(V\) is a function of \(x,y\), then \[ J\left(\frac{\partial^2V}{\partial X^2} + \frac{\partial^2V}{\partial Y^2}\right) = \frac{\partial^2V}{\partial x^2} + \frac{\partial^2V}{\partial y^2}. \]

If \(\lambda = \frac{L_1 x^2 + 2M_1 x + N_1}{L_2 x^2 + 2M_2 x + N_2}\), prove that the condition for \(\lambda\) to attain all real values if \(x\) assumes all real values is \[ (L_1 N_2 - N_1 L_2)^2 < 4(M_1 N_2 - N_1 M_2)(L_1 M_2 - M_1 L_2). \] Find the limitations on the value of \(\lambda\) if this condition is not satisfied. Shew that the turning values of \(\frac{a_1 x^2+b_1}{a_2 x^2+b_2}\) are \(\frac{a_1}{a_2}, \frac{b_1}{b_2}\).

Two men of height \(d\) feet (to the level of the eyes) are walking on the same horizontal level round a conical hill at a depth \(h\) feet below the summit. The total height of the hill is 300 feet and the diameter of its base 800 feet. When the man in front has just disappeared from sight round the hill, shew that the shortest distance \(S\) between them over the hill is given by \[ 5\sin^{-1}\left(\frac{3S}{10h}\right) = 4\cos^{-1}\left(\frac{h-d}{h}\right). \]

\(V\) is the middle point of a given chord \(AB\) of a given circle. \(PQ\) is any parallel chord. \(QV\) meets the circle again in \(R\). Prove that \(PR\) passes through a fixed point.

\(PQ\) is a chord of a parabola that passes through the focus \(S\). Two circles are drawn through \(S\), touching the parabola at \(P\) and \(Q\) respectively. Prove that the circles cut each other orthogonally.

If the tangent at \(P\) to an ellipse meets a directrix in \(R\), and if \(S\) is the corresponding focus, prove that \(PSR\) is a right angle. If two conjugate diameters of an ellipse are produced to meet a directrix, prove that the orthocentre of the triangle so formed is the corresponding focus.

From the point \(Q\) in which the tangent at any point \(P\) of a hyperbola meets an asymptote, perpendiculars \(QM, QN\) are drawn to the axes. Prove that \(MN\) passes through \(P\).

Any number of spheres touch a plane at the same point \(O\). Prove that any plane, not through \(O\), cuts them in a system of co-axal circles.

A variable chord \(PQ\) of a given circle subtends a right angle at a given point \(A\). Find the locus of the pole of \(PQ\) with respect to the circle. Interpret the result when \(A\) lies on the given circle.