Prove that three straight lines in space, parallel to the same plane but not to one another, can be cut by an infinity of straight lines and that the intercepts made by the three lines on any one of the cutting lines are in constant ratio.
Deduce from the focus and directrix definition of an ellipse the existence of a centre and a second focus and directrix. A variable point \(P\) is taken in a line \(AB\) between \(A\) and \(B\): prove that the ellipses of the same fixed eccentricity with \(A, P\) and \(P, B\) respectively as foci intersect on a fixed ellipse of which \(A, B\) are the foci.
Prove that, if \(T\) be a point on a diameter of an ellipse, centre \(C\), and \(V\) be the point in which the polar of \(T\) cuts the diameter, then the rectangle \(CT \cdot CV\) is equal to the square of the semi-diameter. Prove that, as \(T\) varies along the fixed diameter, the locus of the foot of the perpendicular from \(T\) on its polar is a rectangular hyperbola.
Prove that the section of a circular cone by a plane parallel to a tangent plane is a parabola. Deduce that, given a parabola in position, the locus of the vertex of a circular cone, of which the parabola is a section, is another parabola of which the focus and vertex are the vertex and focus respectively of the given parabola.
Prove that the pencil of lines formed by joining any point \(P\) on a circle to four fixed points on the circle has a constant cross ratio: prove also that \(PA, PB\) are harmonic with \(PC, PD\), provided the chords \(AB, CD\) pass each through the pole of the other. Three points \(A, B, C\) lie on a circle and \(A', B', C'\) are chosen on the circle so that \(AA', BC\); \(BB', CA\); and \(CC', AB\) are all harmonic pairs: prove that the lines \(AA', BB', CC'\) are concurrent.
Prove that, if a tangent to a parabola makes an angle \(\theta\) with the axis, the angle \(\phi\) at which the corresponding normal chord cuts the parabola again is given by \[ 2 \tan\theta \tan\phi = 1. \]
Prove that the conditions that the line \(lx+my+n=0\) is respectively a tangent and a normal to the ellipse \(b^2x^2+a^2y^2-a^2b^2=0\) are \[ T \equiv a^2l^2+b^2m^2-n^2=0 \quad \text{and} \quad N \equiv n^2(a^2m^2+b^2l^2)-(a^2-b^2)^2l^2m^2=0. \] Prove also that the quadratic for the tangents (\(t\)) of the angles at which \(lx+my+n=0\) cuts the ellipse is \(t^2N - 2t(a^2-b^2)lmT - (a^2l^2+b^2m^2)T=0\), provided the tangent of the angle at an extremity \((x,y)\) be reckoned as \((b^2mx-a^2ly)/(b^2lx+a^2my)\).
Find the equation of a family of conics, which have a given centre and a given directrix, using the lines of the axes as coordinate axes. Deduce that the envelope of the family consists of two parabolas.
Prove that the locus of the centre of a conic passing through four fixed points is a conic. Show also that, in the special case of conics having triple contact at a point \(P\) and a fourth common point, the curvature of the locus of centres at \(P\) is twice that of any of the conics at \(P\) but is in the opposite sense.
Prove that there are four conics, real or imaginary, with regard to each of which the pair of conics \[ ax^2+by^2+cz^2=0 \quad \text{and} \quad a'x^2+b'y^2+c'z^2=0 \] are polar reciprocal. Show that for \(x^2+y^2-2ax+1=0, x^2+y^2-2bx+1=0\) the four conics are \[ \sqrt{(1+a)(1+b)}(x-1)^2 \pm \sqrt{(1-a)(1-b)}(x+1)^2 \pm 2y^2=0. \] % Note: The signs in the last equation are ambiguous in the source image. The transcribed version reflects the most likely interpretation for generating four conics. The original OCR suggested only plus signs. The scan seems to indicate the first is + and the second and third are \pm. I have used \pm for both for symmetry and to get four conics as the question implies.