Enumerate the principal relations existing between two figures, which are polar reciprocals of one another, distinguishing the case in which reciprocation is with regard to a circle. A conic has a focus at a fixed point \(O\) and has two fixed straight lines as tangents. Shew that their chord of contact passes through a fixed point.
Prove that if \(ABCD\) are fixed points on a conic and \(P\) a variable point then the cross-ratio \(P(ABCD)\) is constant. \(A, B\) are fixed points and \(l\) a fixed straight line. The point \(P\) moves on a conic \(S\) through \(A\) and \(B\). \(PA, PB\) cut \(l\) in \(H, K\) and \(AK, BH\) intersect in \(Q\). Shew that the locus of \(Q\) is a conic \(S'\) and find the condition that \(S\) and \(S'\) coincide.
\(S=0, S'=0, S''=0, S'''=0\) are the equations to four circles. Interpret the equations \(\lambda S + \mu S' = 0\), \(\lambda S + \mu S' + \nu S''=0\), \(\lambda S + \mu S' + \nu S'' + \rho S'''=0\) for different values of \(\lambda, \mu, \nu, \rho\). Shew that the locus of points at which two non-overlapping circles subtend equal angles is a circle coaxal with the given circles.
Find the pole of the line \(lx+my+n=0\) with regard to the conic \(ax^2+by^2=1\), and deduce the tangential equation to the conic. Shew that if two perpendicular straight lines are conjugate with regard to one member of a confocal system of conics they are so with regard to every member, and are tangents to the members which pass through their point of intersection.
State without proof conditions under which the general equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] shall represent (i) a circle, (ii) an ellipse, (iii) a rectangular hyperbola, (iv) a pair of parallel straight lines. A circle touches one straight line and cuts off from another a segment of constant length. Shew that the locus of its centre is a rectangular hyperbola.
Shew that the lines \(Ax^2+2Hxy+By^2=0\) will be conjugate diameters of the conic \(ax^2+2hxy+by^2=1\) if \(aB+bA=2hH\). Determine equations to (i) the axes, (ii) the equi-conjugate diameters.
Shew that the equations \[ \frac{x}{a_1 t^2 + 2b_1 t + c_1} = \frac{y}{a_2 t^2 + 2b_2 t + c_2} = \frac{1}{a_3 t^2 + 2b_3 t + c_3} \] represent a conic with centre given by \[ \frac{x}{a_1 c_3 + a_3 c_1 - 2b_1 b_3} = \frac{y}{a_2 c_3 + a_3 c_2 - 2b_2 b_3} = \frac{1}{2(a_3 c_3 - b_3^2)}. \] Sketch roughly the curve \[ \frac{x}{4t^2} = \frac{y}{(1-t)^2} = \frac{1}{1+t^2}. \]
Two circles intersect in \(A\) and \(B\), any point \(P\) is taken on one of the circles and \(PA\), \(PB\) produced meet the other circle in \(Q, R\); find the envelope of \(QR\).
Describe a circle to pass through two given points and touch (i) a given straight line, (ii) a given circle.
Define the polar of a point with respect to a circle and prove that a straight line is divided harmonically by the point, the circle and the polar of the point. Two chords \(AB, AC\) of a circle are drawn, the perpendicular from the centre on \(AB\) meets \(AC\) in \(D\); prove that the straight line joining \(D\) to the pole of \(BC\) is parallel to \(AB\).