Given the limiting points of a system of coaxal circles, state geometrical constructions for
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 \[ aa'(bc'+b'c)x^2 + bb'(ca'+c'a)y^2 + cc'(ab'+a'b) = 0. \]
The four common points of the parabola given by \(y^2-4ax=0\) and a rectangular hyperbola are all coincident at the point \(P\); prove that the centre of the rectangular hyperbola is the reflexion of \(P\) in the directrix of the parabola and that the asymptotes of the rectangular hyperbola are parallel to the bisectors of the angles between the tangent at \(P\) and the axis of the parabola.
The equation of a conic in homogeneous coordinates is \(s \equiv ax^2+by^2+cz^2=0\), where \(a+b+c=0\); if \(P(f,g,h)\) is a point on this conic \(s\), prove that the conic given by \(afyz+bgzx+chxy=0\) cuts \(s\) in four points \(P, P_1, P_2, P_3\) and that, if \((x_1, y_1, z_1)\) are the coordinates of \(P_1\), the equation of the common chord \(PP_1\) is \(ay_1z_1x + bz_1x_1y+cx_1y_1z=0\). Deduce that the equation of the other common chord \(P_2P_3\) is \(x_1x+y_1y+z_1z=0\) and that the triangle \(P_1P_2P_3\) is self polar with respect to the conic given by \(x^2+y^2+z^2=0\).