If \(L, M\) are the feet of the perpendiculars from the fixed points \(A, B\) respectively to a variable line and \(AL^2 - BM^2 = c^2\), where \(c\) is a constant length, find the envelope of the line and identify this envelope in relation to the points \(A, B\) and the length \(c\).
\(P\) is a point on the circumcircle of the triangle \(ABC\), and the lines through \(P\) perpendicular to \(PA, PB, PC\) meet \(BC, CA, AB\) respectively at the points \(X, Y, Z\). Prove that \(X, Y, Z\) lie on the same diameter of the circumcircle.
If the tangents at the points \(P, Q\) of a parabola meet at \(T\), prove that the circle \(TPQ\) passes through the reflexion of \(T\) in the focus \(S\). If \(P, Q, R\) are the feet of the normals from a point \(N\) to a parabola, prove that the circumcircle of the triangle formed by the tangents to the parabola at \(P, Q, R\) has \(S\) and \(N'\) at the ends of a diameter, where \(N'\) is the reflexion of \(N\) in the focus \(S\).
\(O, P\) are given points on a conic, and a variable pair of lines through \(O\) equally inclined to the chord \(OP\) meet the conic again at the points \(Q, R\); prove that the chord \(QR\) passes through a fixed point \(P'\). If \(O\) is fixed and \(P\) is a variable point on the conic, prove that the locus of the corresponding point \(P'\) is a straight line.
\(l, m\) are two fixed lines in space, which do not lie in the same plane, and \(L, M\) are variable points on \(l, m\) respectively, such that the length \(LM\) is constant. Prove that the plane through \(L\) perpendicular to \(l\) and the plane through \(M\) perpendicular to \(m\) meet in a line, which is a generator of a fixed circular cylinder, whose axis lies along the common normal of the lines \(l, m\).
If \(M, N\) are the feet of the perpendiculars on the coordinate axes from any point \(P\) of the parabola \(y^2 = 4ax\), prove that the perpendicular from \(P\) to \(MN\) is a normal to a fixed parabola, whose equation is of the form \(y^2 = 4b(x+c)\), and find \(b, c\) in terms of \(a\).
\(T\) is a variable point on the line \(lx+my+n=0\), and \(P,Q\) are the points of contact of the tangents from \(T\) to the conic \(ax^2+by^2+c=0\); if the circle on \(PQ\) as diameter meets the conic again at \(P', Q'\), prove that the chord \(P'Q'\) passes through the fixed point, whose coordinates \((x,y)\) are given by \(ax:by:(a+b)c = l:m:(b-a)n\).
Prove that in general two rectangular hyperbolas (real or imaginary) can be found to touch any four coplanar straight lines. If the two rectangular hyperbolas are coincident, prove that their contact points with the four lines are orthocentric, i.e. the line joining any two of them is perpendicular to the line joining the other two.
Prove that, if the equation \(ax^2+by^2+c(x+y+d)^2=0\) (referred to rectangular Cartesian axes) represents an ellipse, the area of the ellipse is equal to \[ \pi abcd^2 (bc+ca+ab)^{-\frac{3}{2}}. \]
A variable conic passes through three given points \(X, Y, Z\) and touches a given line \(p\); prove that the locus of the intersection of the tangents at \(Y, Z\) to this conic is a conic \(\Gamma\), which is inscribed in the triangle \(XYZ\). Prove also that \(p\) is the polar line of \(X\) with respect to \(\Gamma\).