Problems

Prove that the roots of the equation \[ x^4 - x^3\left(4R+2\frac{\Delta}{s}\right) + x^2s^2 + x^2\frac{\Delta}{s}(4R+\frac{\Delta}{s}) - 2s\Delta x + \Delta^2 = 0 \] are the radii of the inscribed and three escribed circles of a triangle whose area is \(\Delta\), circumradius \(R\), and the sum of whose sides is \(2s\).

By considering the expression for \(\cos 7\theta\) in terms of \(\cos\theta\), find the roots expressed in trigonometric form of the equation \[ 64x^6 - 112x^4 + 56x^2 - 7 = 0. \]

Given the limiting points of a system of coaxal circles, state geometrical constructions for

1. the two circles of the system which touch a given line,
2. the circle of the system which passes through a given point,
3. the circle of the system which cuts a given circle orthogonally.

Two fixed lines intersect at the point \(O\), and \(A\) is a fixed point coplanar with them; if a variable circle through the points \(O, A\) meets the fixed lines again at the points \(P, Q\), prove that the envelope of the line \(PQ\) is a parabola with its focus at \(A\) and find the locus of the orthocentre of the triangle \(OPQ\).

Two lines \(l, m\) meet at \(O\) and there is a 1-1 correspondence between the points \(P\) on the line \(l\) and the points \(Q\) on the line \(m\); prove that the locus of the centre of the circle through \(O\) and a pair of corresponding points \(P, Q\) is a hyperbola, whose asymptotes are perpendicular to \(l,m\).

Prove that the pairs of tangents from a fixed point to a pencil of conics touching four fixed lines are in involution. Deduce that there are two parabolas touching the sides of a given triangle \(ABC\) and passing through a given point \(D\), and that, if these parabolas cut orthogonally at \(D\), the four points \(A,B,C,D\) lie on a circle.

A variable sphere passing through a fixed point touches each of two fixed spheres; prove that the locus of each point of contact is a circle.

The equation of the pair of lines \(OA, OB\) referred to rectangular Cartesian axes is \(ax^2+2hxy+by^2=0\); the perpendicular from \((x_1, y_1)\) to \(OA\) meets \(OB\) at \(P\) and the perpendicular from \((x_1,y_1)\) to \(OB\) meets \(OA\) at \(Q\). Prove that the equation of the circle on \(PQ\) as diameter is \[ (a+b)(x^2+y^2) - 2(bx_1 - hy_1)x + 2(hx_1 - ay_1)y + bx_1^2 - 2hx_1y_1 + ay_1^2=0. \]

If a variable chord of the conic given by \(ax^2+2hxy+by^2+2gx+2fy+c=0\) passes through the point \((0,0)\), prove that the locus of the middle point of the chord is given by the equation \(xu+yv=0\), where \[ u = ax+hy+g, \quad v=hx+by+f. \] Identify the locus given by the equation \(xv-yu=0\).

Prove that the eight points of contact of the four common tangents of the conics given by the equations \[ ax^2+by^2+c=0, \quad a'x^2+b'y^2+c'=0 \] lie on the conic given by the equation \[ aa'(bc'+b'c)x^2 + bb'(ca'+c'a)y^2 + cc'(ab'+a'b) = 0. \]