Two particles, of masses \(M\) and \(m\), are connected by an inextensible string of length \(a\). At first \(m\) is held fixed at a height \(a\) above a smooth horizontal plane; the string is initially horizontal and \(M\) is allowed to fall from rest till the string becomes vertical, when \(m\) is let go. Assuming that the string remains taut and that \(M\) is not lifted off the plane, apply the principles of energy and linear momentum to determine the velocities of the particles as \(m\) reaches the plane; prove that \(m\) strikes the plane at an angle \[ \tan^{-1} \frac{\{(2M+m)(M+m)\}^{1/2}}{M}. \]
If a chord of a circle passes through a fixed point within the circle, the rectangle contained by its segments is constant. Through a given point within a circle draw a straight line, the difference of whose segments shall have a given length not exceeding twice the distance of the point from the centre.
Prove that the distances of any point of a circle from a fixed pair of inverse points are in a constant ratio. State any theorems with reference to the inversion of coaxal circles.
When is a pencil of rays said to be in involution? Shew that if two conjugate rays intersect at right angles then all conjugate pairs do so. Shew also that, if three rays intersect their conjugates at the same angle, that angle is a right angle.
Give and justify geometrical constructions
Find the coordinates of the pole of the line \(lx+my=1\) with regard to the parabola \(y^2 = 4ax\). Prove that, if the pole of one of the lines \[ y-2a = \lambda(x+2a), \quad y+2a = \mu(x+2a), \] lies on the other, they intersect on the hyperbola \(2y^2 - x^2 = 4a^2\).
Two normal chords of a parabola make angles with the axis whose cosines are \(\frac{1}{3}\) and \(\frac{2}{3}\) respectively. Prove that their lengths are equal. What is the minimum length of a normal chord?
Prove that when a circle intersects an ellipse their common chords are equally inclined to the axes. If the circle passes through the centre of the ellipse, and one of a pair of common chords passes through a fixed point on the major axis, the perpendicular from the centre on the other chord will meet it on a fixed ordinate.
A rectangular hyperbola circumscribes a fixed right-angled triangle. Shew that its centre lies on a fixed circle passing through the right angle.
Find in trilinear coordinates the equation of the circle which has for its diameter the perpendicular drawn from the angular point \(A\) of the triangle of reference to the side \(BC\). Find also the radical axis of this circle and the circumscribed circle.