Problems

The points in a plane are displaced so that the point \((x,y)\) referred to rectangular coordinates takes the position \((X,Y)\), where \(X=px+qy, Y=rx+sy\). Show that a unit square in any position becomes a parallelogram of area \(ps \sim qr\), and that the parallelogram has the sum of the squares of the lengths of its sides constant. What is the least possible angle between the sides of the parallelogram?

The three sides of a varying triangle touch the parabola \(y^2=4ax\), and two of the vertices lie on the confocal parabola \(y^2=4(a+\lambda)(x+\lambda)\); prove that the third vertex lies on the confocal \(y^2=4(a+\mu)(x+\mu)\), where \(a\mu=4\lambda(a+\lambda)\).

Show that the focal radius vector \(r\) of a point on an ellipse, the angle \(\theta\) made by the vector with the major axis and the eccentric angle \(\phi\) of the point are connected by the relations \[ \frac{l}{r} = 1+e\cos\theta, \quad r=a(1-e\cos\phi), \quad \tan\frac{\theta}{2} = \sqrt{\frac{1+e}{1-e}}\tan\frac{\phi}{2}. \] Show also that the maximum value of \(\theta-\phi\) is \(2\sin^{-1}\sqrt{\frac{a-b}{a+b}}\), and that it occurs when \(\theta+\phi=\pi, r=b\).

Find the equation to the pair of tangents drawn from a point to the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-1=0\). A pair of tangents to any confocal of \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-1=0\) pass respectively through the fixed points \((0,c_1)\) and \((0,c_2)\): show that the intersection of the tangents lies on the circle \[ (x^2+y^2-a^2+b^2)(c_1+c_2) = 2y(c_1c_2-a^2+b^2). \]

Find the condition that the line \(lx+my+n=0\) touches the conic \[ ax^2+by^2+c+2fy+2gx+2hxy=0. \] Show that, if \(x\cos\psi+y\sin\psi=p_1\), \(x\cos\psi+y\sin\psi=p_2\) represent a variable pair of parallel tangents of a fixed conic, the lines \[ x\cos\psi+y\sin\psi=\lambda p_1 + \mu p_2, \quad x\cos\psi+y\sin\psi=\lambda p_2 + \mu p_1, \] where \(\lambda, \mu\) are constants, envelope another fixed conic with parallel asymptotes.

Show that the equation of a conic may be put in the form \(zx-y^2=0\), when homogeneous coordinates are used, and that the triangle of reference may be chosen in a doubly infinite number of ways. Given \(Y=x-(a+b)y+abz\), determine \(X\) and \(Z\), so that \(ZX-Y^2=0\) and \(zx-y^2=0\) may represent the same conic.

Resolve into factors:

1. \((bc+ca+ab)^3 - abc(a+b+c)^3\);
2. \((bcd+cda+dab+abc)^2 - abcd(a+b+c+d)^2\).

Find the condition that \[ (b-c)^n + (c-a)^n + (a-b)^n \] should be divisible by \(\Sigma a^2 - \Sigma bc\).

Find the conditions that

1. \(ax^2+2bx+c\) may be positive for all real values of \(x\);
2. \(ax^2+2hxy+by^2+2gx+2fy+c\) may be positive for all real values of \(x\) and \(y\).

Sum the series:

1. \(\dfrac{1}{1.2} + \dfrac{1}{2.3} + \dfrac{1}{3.4} + \dots\) to \(n\) terms;
2. \(1^2+2^2x+4^2x^2+5^2x^3+7^2x^4+8^2x^5+\dots\) to infinity.

Prove that \(\log(1+x)=x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dots\), if \(|x|<1\). Prove that \((1+x)^{1/x} = e - \dfrac{ex}{2} + \dfrac{11e}{24}x^2 - \dfrac{7e}{16}x^3 + \text{etc.}\)