\(Q\) and \(R\) are the inverse points of \(P\) with respect to two fixed circles. Prove that, when \(P\) moves on a circle coaxal with the fixed circles, the line \(QR\) passes through a fixed point.
Prove that the circumcircle of a triangle passes through the focus of any parabola which touches its sides. From any point \(T\) on a given diameter of a parabola, whose focus is \(S\), tangents \(TP, TP'\) are drawn to the curve. Prove that the centre of the circle \(TPP'\) lies on a fixed straight line perpendicular to the axis and that its radius is proportional to \(ST\).
From a fixed point \(T\) two tangents are drawn to any one of a system of confocal ellipses. Prove that the sum of the angles which the line joining the foci subtends at the two points of contact is constant.
Determine the centre and the radius of the circle inscribed in the triangle formed by the lines \(3x+4y=0\), \(12x+5y=0\), \(5x+12y-17=0\).
Show how to construct the common normal of any two lines in space and prove that in general it is unique.
\(H, H'\) are two points on the major axis and \(K, K'\) two points on the minor axis of an ellipse, such that the four normals at the ends of the chords lying along \(HK, H'K'\) are concurrent in a point \(P\). Prove that the four normals at the ends of the chords lying along \(HK', H'K\) are concurrent in a point which lies on the diameter through \(P\).
Prove that the axes of the conic \(ax^2+2hxy+by^2+2gx+2fy+c=0\) are given by \(h(\xi^2-\eta^2)-(a-b)\xi\eta=0\), and its asymptotes by \(b\xi^2-2h\xi\eta+a\eta^2=0\), where \(\xi = ax+hy+g, \eta=hx+by+f\).
Find the foci of the conic whose tangential equation is \[ Al^2+2Hlm+Bm^2+Cn^2=0. \] Hence, or otherwise, determine the real foci and the eccentricity of the conic whose locus equation is \(16x^2-4xy+19y^2-300=0\).
Prove that, if \(u=0, v=0\) and \(u'=0, v'=0\) are the equations of two pairs of conjugate diameters of a conic, the equation of any pair is \(uv+\lambda u'v'=0\), where \(\lambda\) is a parameter. A conic circumscribes the triangle of reference and has its centre at the point \((x', y', z')\). Shew that the equation of any pair of conjugate diameters can be written in the form \[ (\beta-\gamma)(\alpha'y-y'z)^2+(\gamma-\alpha)(x'z-z'x)^2+(\alpha-\beta)(y'x-x'y)^2=0, \] the coordinates being areal.
If \(\alpha\) is a root of \(ax^2+2bx+c=0\) and \(\beta\) a root of \(a'x^2+2b'x+c'=0\), find the equation whose roots are the different values of \(\alpha/\beta\).