Find equations for the incentre of the triangle formed by the lines \[ x-2y=0, \quad 4x-3y+5=0, \quad x+3y-10=0, \] and obtain the coordinates of the orthocentre.
A line \(l\) meets the parabola \(y^2=4ax\) in \(P\) and \(Q\). The line through \(P\) parallel to the tangent at \(Q\) meets the line through \(Q\) parallel to the tangent at \(P\) in a point \(R\). If \(l\) varies, passing through a fixed point \((\alpha, \beta)\), shew that the locus of \(R\) is the parabola \[ 2y^2 - 2ax - 3\beta y + 6a\alpha = 0. \]
Shew that the equations of the chords of contact of any conic \(S\) which has double contact with each of two given conics \(S_1\) and \(S_2\) are of the form \(\lambda\alpha+\mu\beta=0\) and \(\lambda\alpha-\mu\beta=0\), where \(\alpha=0\) and \(\beta=0\) are lines joining the points of intersection of \(S_1\) and \(S_2\) in pairs. Conversely for any value of \(\frac{\lambda}{\mu}\) a conic can be found having double contact with \(S_1\) along \(\lambda\alpha+\mu\beta=0\) and with \(S_2\) along \(\lambda\alpha-\mu\beta=0\). Shew that there are six conics \(S\) which pass through a given point, and six conics \(S\) such that the chords of contact are conjugate with respect to \(S\).
Shew that the locus of poles of a line \(l\) with respect to the conics of a confocal system is a line perpendicular to \(l\), and that this is the normal to the conic of the system which touches \(l\) at its point of contact with \(l\). Hence shew that if tangents are drawn from \((x_1, y_1)\) to the conic \[ \frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda} = 1, \] the normals at the points of contact will intersect at the point \((\xi, \eta)\) given by \[ \frac{\xi}{x_1(b^2+\lambda-y_1^2)} = \frac{-\eta}{y_1(a^2+\lambda-x_1^2)} = \frac{a^2-b^2}{(a^2+\lambda)y_1^2 + (b^2+\lambda)x_1^2}. \] Deduce that the locus of \((\xi, \eta)\) as \(\lambda\) varies is a straight line.
Find the equation of the line \(l\) joining the points of intersection of \(x = \lambda y\) and \(x = \mu z\) with the conic \[ ayz+bzx+cxy=0 \] other than the vertices of the triangle of reference. If \(\lambda\) and \(\mu\) are allowed to vary subject to the condition \[ a\lambda\mu + \beta\lambda + \gamma\mu + \delta = 0, \] shew that the line \(l\) will pass through a fixed point provided that \(a\alpha = c\beta+b\gamma\), and find the coordinates of the point.
A straight uniform rod of length \(2l\) rests in contact with a small smooth fixed peg, the lower end of the rod being on a smooth plane which is inclined at an angle \(\alpha\) to the horizontal. If \(c\) is the shortest distance of the peg from the plane, and \(\theta\) the angle the rod makes with the vertical, prove that \[ l \sin\theta \cdot \cos^2(\theta-\alpha) = c\sin\alpha. \]
Three uniform freely jointed rods form an isosceles triangle \(ABC\). \(P\) is the weight of each of the equal rods \(AB, AC\), and \(Q\) is the weight of \(BC\). The figure is suspended from \(A\). Prove that the reaction between the rods at either of the lower hinges makes with the vertical an angle whose tangent is \(\displaystyle\left(1+\frac{Q}{2P}\right)\tan\frac{A}{2}\).
Prove that the greatest inclination to the horizontal at which a uniform rod can rest inside a rough sphere is given by \(\tan\theta = \displaystyle\frac{\sin\lambda.\cos\lambda}{\cos^2\alpha - \sin^2\lambda}\), where \(\lambda\) is the angle of friction, and the rod subtends an angle \(2\alpha\) at the centre of the sphere.
A fine elastic string \(OAB\), whose modulus of elasticity is \(\lambda\) and unstretched length is \(a\), has one end fixed at \(O\), and passes over a small smooth fixed peg at \(A\), where \(OA=a\). A particle of mass \(m\) hangs in equilibrium at \(B\). Prove that if a horizontal impulse \(I\) is applied to the particle, it will move in a horizontal straight line with simple harmonic motion of amplitude \(I\left(\displaystyle\frac{a}{\lambda m}\right)^{\frac{1}{2}}\).
Four heavy particles lie in a straight line on a smooth horizontal plane. The first is projected along the straight line to strike the second, which in turn strikes the third, which in turn strikes the fourth. The first three particles are now at rest, and the coefficient of restitution at each impact was \(e\). Prove that the final energy is \(e^3\) times the energy just before the first impact.