Problems

State and prove the harmonic property of a quadrangle. If \(L, M, N\) are the feet of the perpendiculars from the vertices \(A, B, C\) of a triangle to the opposite sides, prove that the triangle \(LMN\) is self polar with respect to any rectangular hyperbola through \(A, B, C\).

Prove that the equation of the circumcircle of the triangle whose sides lie along the lines \(ax^2+2hxy+by^2=0\), \(lx+my+n=0\) is \[ (am^2-2hlm+bl^2)(x^2+y^2) + n(a-b)(my-lx) - 2hn(mx+ly)=0. \] Interpret this result geometrically, when \(am^2-2hlm+bl^2=0\).

The sides of a variable triangle with its centroid at the fixed point \((x_1, y_1)\) touch the parabola \(y^2=4ax\); prove that the vertices of the triangle lie on the parabola \[ 2(y^2-ax)-3y_1y+6ax_1=0. \]

Prove that the condition that the line \(lx+my+n=0\) should touch the parabola, whose focus is \((\alpha, \beta)\) and whose directrix is \(px+qy+r=0\), is \[ (p\alpha+q\beta+r)(l^2+m^2) - 2(pl+qm)(\alpha l + \beta m + n) = 0. \]

Prove that the locus of the poles of a given straight line with respect to a system of confocal conics is a straight line. If the given straight line is \(lx+my+n=0\) and one of the conics is \(ax^2+2hxy+by^2+c=0\), prove that the locus of poles is the line \[ n(ab-h^2)(mx-ly) + c\{(a-b)lm - h(l^2-m^2)\} = 0. \] Interpret this result geometrically, when (i) \((a-b)lm-h(l^2-m^2)=0\), (ii) \(ab-h^2=0\).

A variable conic touches a fixed line and also touches the sides of a fixed triangle; prove that for any such conic the lines joining the vertices of the triangle to the points of contact of the opposite sides meet in a point \(P\) and that, as the conic varies, the locus of \(P\) is a conic circumscribing the fixed triangle.

If \(a,b\) and \(c\) are all positive, show that \[ 3(a^3+b^3+c^3) \ge (a^2+b^2+c^2)(a+b+c). \] Hence or otherwise, show that \[ \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \ge \frac{3}{2}. \]

Find the cubic, with unity as the coefficient of the highest term, which has the roots \[ 2\cos\frac{2\pi}{7}, \quad 2\cos\frac{4\pi}{7}, \quad 2\cos\frac{6\pi}{7}. \]

A system of conics is such that all the conics have a common focus and touch each of two parallel lines. Prove that the directrices corresponding to the common focus are concurrent, that the centres are collinear, and that the envelope of the asymptotes is a circle with its centre at the common focus.

Find the condition that the line joining the points \((t_1^2, t_1, 1)\), \((t_2^2, t_2, 1)\) on the conic \[ S \equiv y^2-zx=0 \] should meet the conic \[ S' \equiv ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0 \] in a pair of points conjugate with respect to \(S\). Hence find the envelope of lines which meet \(S\) and \(S'\) in pairs of points which harmonically separate each other. \(S\) and \(S'\) intersect at \(A, B, C, D\). Show that the eight tangents to \(S\) and \(S'\) at \(A, B, C, D\) all touch the envelope.