Put into real partial fractions
Prove that with the usual notation \[ Rr = \frac{abc}{4s} \quad \text{and} \quad r_1+r_2+r_3=r+4R. \] Shew that the radius of the inscribed circle of a triangle whose sides are \(a+x\), \(b+x\) and \(c+x\), where \(x\) is small, is approximately \[ r + \frac{2R-r}{2s}x. \]
Prove that, when \(n\) is a positive integer, \[ \tan n\theta = \frac{n\tan\theta - \frac{1}{3!}n(n-1)(n-2)\tan^3\theta+\dots}{1-\frac{1}{2!}n(n-1)\tan^2\theta+\frac{1}{4!}n(n-1)(n-2)(n-3)\tan^4\theta-\dots}. \] Find an equation whose roots are the tangents of \(\theta, 2\theta, 4\theta, 5\theta, 7\theta\) and \(8\theta\) where \(\theta=20^\circ\), and shew that \(\tan 20^\circ \tan 40^\circ \tan 80^\circ = \sqrt{3}\).
The conic \(ax^2+2hxy+by^2+2gx+2fy+c=0\) is cut by the straight line \(lx+my+1=0\) in \(P\) and \(Q\). Shew that the area of the triangle \(OPQ\), where \(O\) is the origin, is \[ \frac{\{-\left(Al^2+2Hlm+Bm^2+2Gl+2Fm+C\right)\}^{\frac{1}{2}}}{am^2-2hlm+bl^2}, \] where \(A, B, C, F, G, H\) are the minors of \(a, b, c, f, g, h\) in \[ \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}. \]
A family of conics circumscribe the triangle \(ABC\) and pass through its centroid \(G\). Tangents to one of these conics at \(A, B, C\) cut the opposite sides in \(D, E, F\). Shew that the locus of the centroid of the triangle \(DEF\) is a cubic curve passing through \(G\) and touching the sides of the triangle \(ABC\) at their middle points.
Find the area of a loop of the curve \(y^2=x^2-x^4\). \par Find also the distance from the origin of the centre of gravity of the area included within the loop.
\(ABCD\) is a square formed of four light rods jointed together, the diagonal \(AC\) being a fifth light rod. Weights \(P\) and \(Q\) are attached to the corners \(B, D\) respectively, and the system is hung up by the corner \(A\). Find the inclination to the vertical of the rod \(AC\) and also the stress in it.
A train weighs 200 tons and the engine exerts a constant pull of 45 lb. per ton, resistance to motion being 10 lb. per ton. The train starts from rest; after a certain time steam is turned off and the brakes put on. The train comes to rest at a distance of 1050 yards from the starting point 2 mins. 20 secs. after it started. Find the retarding force per ton of the brakes, and also the greatest horse-power developed.
A particle starts from rest at any point \(P\) in the arc of a smooth cycloid whose axis vertical and vertex \(A\) downwards; prove that the time of descent to the vertex is \(\pi\sqrt{\frac{a}{g}}\), where \(a\) is the radius of the generating circle. \par Shew also that if the particle is projected from \(P\) downwards along the curve with velocity equal to that with which it reaches \(A\) when starting from rest at \(P\), it will now reach \(A\) in half the time taken in the preceding case.
Shew that, if \[ \frac{x^2}{a} + \frac{y^2}{b} = x+y \quad \text{and} \quad \frac{a^2}{x} + \frac{b^2}{y} = a+b, \] either \[ \frac{x}{a} + \frac{y}{b} + 1 = 0 \quad \text{or} \quad x+y=a+b. \]