Shew that there are four normals from a point \((h,k)\) to the ellipse \(\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\); that the feet of these normals lie on a rectangular hyperbola passing through the centre of the ellipse; that two of these feet are always real points, one lying in the same quadrant with \(h,k\) and the other in the opposite quadrant, and that the other two, if real, are on the same side of the axis of \(y\) as the point \((h,k)\), and on the opposite side of the axis of \(x\). Shew that the normals at the four points at which the ellipse is met by the lines \[ \frac{mx}{a} + \frac{ny}{b} = 1, \quad \frac{x}{ma} + \frac{y}{nb} = -1 \] are concurrent in the point given by \[ \frac{ah}{a^2-b^2} = \frac{-n+1/n}{m/n+n/m}, \quad \frac{bk}{a^2-b^2} = \frac{m-1/m}{m/n+n/m}. \]
If four points on a rectangular hyperbola are such that the chord joining any two is perpendicular to the chord joining the other two shew that the same is true for the other pairs of chords joining the points. Shew also that the three pairs of chords joining any four points on a hyperbola cut off on either asymptote three segments which have a common middle point, and that all conics through the four points cut off segments from the asymptote with the same middle point.
Find the equation of the conic given by \[ x:y:1 = S_1(t):S_2(t):S_3(t), \] where \begin{align*} S_1(t) &= a_1t^2+b_1t+c_1, \\ S_2(t) &= a_2t^2+b_2t+c_2, \\ S_3(t) &= a_3t^2+b_3t+c_3. \end{align*} Shew that one of the asymptotes of the conic has the equation \[ xS_3(\alpha) - yS_1(\alpha) = (C\alpha^2-2B\alpha+A)/(\alpha-\beta)a_3, \] where \(\alpha, \beta\) are the roots of \(S_3(t)=0\), and \[ A=b_1c_2-b_2c_1, \quad B=c_1a_2-a_1c_2, \quad C=a_1b_2-a_2b_1, \] the other asymptote being obtained by interchanging \(\alpha\) and \(\beta\).
Shew that the tangential equation of all conics having the real points \((a,b)\) \((a',b')\) for foci is \[ (la+mb+1)(la'+mb'+1) - \lambda(l^2+m^2) = 0 \] and obtain the corresponding point equation. Shew that the hyperbolas of the system are given by negative values of \(\lambda\), and the ellipses by positive values.
Prove that:
Express \(1-\cos^2\theta-\cos^2\phi-\cos^2\psi-2\cos\theta\cos\phi\cos\psi\) as a product of four cosines. Eliminate \(\theta\) from \(a\sin\theta+b\cos\theta=2c\cos2\theta\), \(a\cos\theta-b\sin\theta=c\sin2\theta\).
\(ABCD\) is a quadrilateral circumscribing a circle and \(a,b,c,d\) are the lengths of the tangents from \(A,B,C,D\) respectively; prove that the sum of a pair of opposite angles is \(2\theta\), where \((a+b)(b+c)(c+d)(d+a)\cos^2\theta=(ac-bd)^2\).
Express \((a+ib)^{c+id}\) in the form \(A+iB\) where \(i=\sqrt{-1}\). If \(\sin x = y\cos(x+a)\), expand \(x\) in ascending powers of \(y\).
State the laws of friction. On the radius \(OA\) of a circular disc as diameter a circle is described, and the disc enclosed by it is cut out. If the remaining solid rest in a vertical plane on two rough pegs in a horizontal plane subtending an angle \(2\alpha\) at the centre \(O\), show that the greatest angle that \(OA\) can make with the vertical is \(\sin^{-1}(3\sin2\lambda\sec\alpha)\), where \(\lambda\) is the angle of friction at the pegs.
State the principle of virtual work and prove it in the case of a single lamina acted on by forces in its plane. \(ABCD\) is a rhombus formed by four light rods smoothly jointed at their ends and \(PQ\) is a light rod smoothly jointed at one end to a point \(P\) in \(BC\) and at the other end to a point \(Q\) in \(AD\). Two forces each equal to \(F\) are applied at \(A\) and \(C\) in opposite directions along \(AC\). Prove that the stress in \(PQ\) is \(F.AB.PQ/AC(AQ\sim BP)\).