Define the radical axis of two circles and prove that the difference of the squares of the tangents from a point to two circles varies as the distance of the point from their radical axis.
Prove that if two planes be each perpendicular to another plane, their line of section is perpendicular to this plane. In a tetrahedron \(ABCD\) if \(AB\) is equal to \(CD\) and \(AC\) is equal to \(BD\), shew that the line which joins the middle points of \(AD\) and \(BC\) is perpendicular to both these edges.
Define a range of points in involution; and prove that if \(\{AA', BB', CC', \dots\}\) be such a range, the cross ratios of the ranges \(\{AA'BC'\}\), \(\{A'AB'C'\}\) are equal. If \(O\{AA', BB', CC'\}\) be a pencil in involution, and if a circle be drawn through \(O\) cutting the rays in \(A, A'\), etc., shew that the chords \(AA', BB', CC'\) are concurrent.
If \(PT, PT'\) are tangents to an ellipse of which \(S, S'\) are foci, prove that the angles \(SPT\) and \(S'PT'\) are equal. If \(P\) lies on the auxiliary circle prove that \(ST\) and \(S'T'\) are parallel.
Find the area of a triangle, the coordinates of whose angular points are given. \(A, B, C, D\) are four fixed points, prove that the locus of a point \(P\) which moves so that (i) the sum, (ii) the difference of the areas of the triangles \(PAB\) and \(PCD\) is constant is a straight line in each case.
Find the length of the tangent from a given external point to the circle \[ x^2+y^2+2gx+2fy+c=0. \] Shew that the equation of the circle on the chord \(x\cos\alpha+y\sin\alpha-p=0\) of the circle \(x^2+y^2-a^2=0\) as diameter is \[ x^2+y^2-a^2 - 2p(x\cos\alpha+y\sin\alpha-p)=0. \]
Find the equation of the normal to the parabola \(y^2=4ax\), which makes an angle \(\phi\) with the axis of \(x\). Prove that the length of this normal chord intercepted by the curve is \(4a\sec\phi\operatorname{cosec}^2\phi\). Also prove that the locus of the middle points of normal chords is \[ y^4 - 2axy^2 + 4a^2y^2 + 8a^4 = 0. \]
Find the equation of the pair of tangents drawn from a given point to the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Two tangents are drawn from a point \(P\) to a conic to meet a fixed tangent at \(Q\) in \(T\) and \(T'\). Shew that if \(QT \cdot QT'\) is constant the locus of \(P\) is a straight line parallel to the tangent at \(Q\).
A point on the conic \(y^2=kxz\) is given by the parameter \(\lambda\) where \(x=\lambda^2y\); prove that the equation of the line joining the points \(\lambda\) and \(\lambda'\) is \[ x-y(\lambda+\lambda') + k\lambda\lambda'z = 0. \] Each of two sides of a triangle inscribed in a conic passes through a fixed point; prove that the third side always touches a conic having double contact with the given one.
Factorise