10273 problems found
If the normals to the conic \(ax^2+by^2+c=0\) at the ends of the chord \(ahx+bky+c=0\) meet at \(P\), prove that two other normals to the conic pass through \(P\) and that the equation of the chord through the feet of these normals is \(kx+hy+hk=0\). If \(ah^2+bk^2+c=0\), deduce the coordinates of the centre of curvature at the point \((h,k)\) of the conic.
The coordinates of a variable point are \(x=(3t+1)/(t+1)\), \(y=2t/(t-1)\), where \(t\) is a parameter; prove that the locus of the point is a rectangular hyperbola and find the equation of (i) the tangent at the point with parameter \(t\), (ii) each of its asymptotes. Prove that, if the four points given by \(t=a,b,c,d\) lie on a circle, \[ a+b+c+d+bcd+cad+abd+abc=0. \]
Prove that with a proper choice of homogeneous coordinates the equation of a variable conic through four fixed points can be put into the form \(ax^2+by^2+cz^2=0\), where \(a:b:c\) are parameters and \(a+b+c=0\). Prove that two of the conics through the four fixed points touch the line \(lx+my+nz=0\), and find the coordinates of their points of contact.
\(A_1A_2A_3A_4A_5A_6A_7\) is a regular heptagon, and the lines \(A_2A_5, A_2A_6, A_2A_7\) meet \(A_1A_4\) in \(B_5, B_6, B_7\) respectively. Prove that
Five points in a plane are given, no three of them lying on a straight line. Prove that at least one of the quadrangles determined by a set of four out of the five points is convex.
Express the polynomials \(x^8-34x^4+1\), \(x^8+34x^4+1\) as the product of irreducible polynomials with integer coefficients.
Prove that the radius of curvature of the envelope of the line \[ x\cos\theta+y\sin\theta+f(\theta)=0 \] at its point of contact with the line is \(\pm[f(\theta)+f''(\theta)]\). Deduce that a circle of radius \(a (\ne 0)\) is the only curve whose radius of curvature at every point is equal to \(a\).
If \(x\) is any complex root of the equation \(x^{11}-1=0\), and if \[ a=x+x^3+x^4+x^5+x^9, \quad b=x^2+x^6+x^7+x^8+x^{10}, \] prove that \((a-b)^2 = -11\). Show further that \[ (x^3+1)[a-b-2(x-x^{10})] = x^3-1, \] and deduce that \[ \tan\frac{3\pi}{11} + 4\sin\frac{2\pi}{11} = \sqrt{11}. \]
If \[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}(x^2-1)^n, \] prove that \[ \int_{-1}^1 P_m(x)P_n(x) \, dx = 0 \text{ if } m \ne n \] \[ = \frac{2}{2n+1} \text{ if } m=n. \]
The diagram represents a girder bridge in which the horizontal and vertical girders are of equal length, and the remaining girders are inclined at an angle of 45\(^\circ\) to the horizontal. The bridge rests on smooth horizontal supports at \(A\) and \(G\) and carries loads at \(B, C, D, E, F\) as indicated. The weights of the girders may be neglected, and all the joints are assumed to be smooth. [A diagram shows a Warren truss with vertical members. A is the left support, G is the right support. Top chord nodes are H, K, L, M, N. Bottom chord nodes are A, B, C, D, E, F, G. The bays are square. From left to right, loads are 4W at B, 3W at C, 4W at D, W at E, 2W at F.] Draw the force diagram, and deduce from it the stress in the girder \(LD\). Indicate in which of the girders the stress due to the loads is a tension, and in which it is a thrust.