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.
Prove that the parallels to the sides of a triangle drawn through any point cut the sides in six points which lie on a conic.
Having given the centre of a conic and three tangents, shew how to construct any number of other tangents and their points of contact.
If the coordinates of any point referred to two different sets of axes (not necessarily rectangular) are connected by the relations \(x=a\xi+b\eta+c\), \(y=a'\xi+b'\eta+c'\), prove that \((ab'-a'b)(a'b-ab') = aa'-bb'\).
Prove that, if the sum of the inclinations to the axis of \(x\) of normals drawn from the point \((x,y)\) to the ellipse \(b^2x^2+a^2y^2=a^2b^2\) is an odd multiple of a right angle, then the locus of \((x,y)\) is \(x^2-y^2=a^2-b^2\).
A triangle is inscribed in the conic \(x^2+y^2+z^2=0\), and two of its sides touch the conic \(ax^2+by^2+cz^2=0\). Shew that the envelope of the third side is \[ (-bc+ca+ab)^2x^2 + (bc-ca+ab)^2y^2 + (bc+ca-ab)^2z^2=0. \]
Prove that \[ 10^n - (5+\sqrt{17})^n - (5-\sqrt{17})^n \] is divisible by \(2^{n+1}\).