\(AOA'\) is a fixed diameter of an ellipse whose centre is \(O\), and \(P,Q\) are points in which the ellipse is cut by a pair of conjugate diameters. Prove by orthogonal projection or otherwise that the locus of the point of intersection of \(AP\) with \(A'Q\) is a similar ellipse whose centre lies on the given ellipse.
Prove that \(a+b+c+d\) is a factor of the expression \[ 2(a^4+b^4+c^4+d^4)-(a^2+b^2+c^2+d^2)^2+8abcd. \] Shew that \(a+b-c-d\) is also a factor, and find the remaining factors.
Shew that, if \(b\) is small compared with \(a\), the expression \((a-b)^n/(a+b)^n\) is approximately equal to \((a-nb)/(a+nb)\). When \(n<\frac{1}{2}\), and \(b/a < 10^{-2}\), determine the degree of approximation.
Prove that \[ \cos 7x - 8\cos^7x = 7\cos x\cos 2x\left(\cos 2x - 2\cos\frac{\pi}{5}\right)\left(\cos 2x - 2\cos\frac{3\pi}{5}\right). \]
The sides of a parallelogram are \(a\) and \(b\) and the acute angle between them is \(\alpha\); the acute angle between the diagonals is \(\theta\). Prove that \[ (a^2-b^2)\tan\theta = 2ab\sin\alpha. \] Determine the greatest value of the acute angle of a parallelogram whose diagonals have given lengths \(p\) and \(q\).
A variable line passes through a fixed point \((a,b)\) and cuts the co-ordinate axes in \(H\) and \(K\). The lines drawn through \(H\) parallel and perpendicular to a given line \(y=mx\) cut the axis \(x=0\) in \(Y\) and \(Y'\); and the lines drawn through \(K\) parallel to \(HY\) and \(HY'\) cut the axis \(y=0\) in \(X\) and \(X'\). Shew that the lines \(XY'\) and \(X'Y\) each pass through a fixed point which lies on a line that passes through the origin and is independent of \(m\).
Prove that the line drawn through any point of the parabola \(y^2=4ax\) at right angles to the line joining the point to the vertex is normal to a fixed parabola whose equation is of the form \(y^2=16a(x+4a)\).
The line \(y=k\) cuts the ellipse \(b^2x^2+a^2y^2=a^2b^2\) in \(K\) and \(K'\); through these points any parallel lines \(KP, K'P'\) are drawn cutting the ellipse in \(P\) and \(P'\). Prove that the locus of the pole of \(PP'\) is a similar co-axal ellipse.
Prove that, if \(P\) and \(Q\) are points on the cardioid \(r=a(1+\cos\theta)\) such that the angle between the tangents at \(P\) and \(Q = \alpha\), the chord \(PQ\) subtends an angle \(\frac{1}{2}(\pi-\alpha)\) at the cusp.
A frame consists of seven light rods jointed to form three equilateral triangles \(ABC, BCD, CDE\). The frame rests on smooth vertical supports at \(A\) and \(E\), with \(ACE\) and \(BD\) horizontal, \(BD\) being above \(AE\), and carries loads of 12 cwt. at \(B\) and 10 cwt. at \(C\). Determine the stresses in the rods, stating which are in tension and which in compression.