Problems

Normals are drawn at the extremities of any chord passing through a given fixed point on the axis of the parabola. Prove that the locus of their point of intersection is a parabola.

Find the equation of the polar of \((h,k)\) with respect to the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \] Show that the equations of a pair of perpendicular lines, each of which passes through the pole of the other, may be written \[ lx+my+n=0, \quad n(mx-ly)+lm(a^2-b^2)=0. \] Show also that the product of the distances of such a pair of lines from the centre depends only on their directions and cannot exceed \(\frac{1}{2}(a^2-b^2)\).

Find an equation whose roots are the squares of the semi-axes of the conic \[ ax^2+2hxy+by^2=1. \] Prove also that the equation of the equal conjugate diameters of the conic is \[ \frac{ax^2+2hxy+by^2}{ab-h^2} = \frac{2(x^2+y^2)}{a+b}. \]

If two conics have each double contact with a third conic, prove that their chords of contact with the third conic, and a pair of their chords of intersection with each other, will all meet in a point and form a harmonic pencil. Prove also that if any three conics are drawn all passing through two given points A and B, their three common chords that do not pass through A or B are concurrent.

Prove that, in areal co-ordinates, the equation of an asymptote of the conic \[ yz=kx^2 \] is \[ 2k\mu x = ky + \mu^2 z, \] where \(\mu\) is given by the equation \[ \mu^2+\mu+k=0. \] Prove also that the asymptotes, for various values of \(k\), envelope a parabola whose equation is \[ (y-z)^2+4x(x+y+z)=0. \]

Solve the equations:

1. [(i)] \(16x(x+1)(x+2)(x+3)=9\),
2. [(ii)] \begin{align*} x+y+z &= axyz, \\ yz+zx+xy&=-b, \\ (1+x^2)(1+y^2)(1+z^2)&=(1+b)^2, \end{align*} where \(a\) is not equal to 1.

If \(|x|<1\), sum to infinity the series whose \(n\)th terms are

1. [(i)] \(n^2 x^n\),
2. [(ii)] \(\log(1+x^{2^n})\).

Prove that \[ \begin{vmatrix} (b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2 \end{vmatrix} = 2abc(a+b+c)^3. \]

(i) If \(c_r\) is the coefficient of \(x^r\) in the expansion of \((1+x)^n\) in a series of ascending powers of \(x\), prove that \[ c_1^2+2c_2^2+3c_3^2+\dots+nc_n^2 = \frac{2n-1!}{n-1!n-1!}. \] (ii) If \(a\) and \(b\) are unequal, and if \(\frac{a-bx}{(1-x)^2}\) is equal to the sum of the first \(r\) terms of its expansion in a series of ascending powers of \(x\), prove that \[ x=\frac{a+r(a-b)}{b+r(a-b)}. \]

Prove that, if \(a\) and \(b\) are positive integers,

1. \(\frac{a^5}{120} - \frac{a^3}{24} + \frac{a}{30}\) is an integer,
2. the probability that \(\frac{1}{5}(a^2+b^2)\) is an integer is \(\frac{9}{25}\).