If \(S=0\) be the equation of a circle and \(\alpha=0, \beta=0\) are the equations of straight lines, assign geometrical meanings to the expressions \(S, \alpha, \beta\). \par Also interpret the equations \[ S-\kappa\alpha\beta=0, \quad S-\kappa\alpha^2=0 \] and state what geometrical theorems they represent.
Find the condition that the four lines given by the equations \[ ax^2+2hxy+by^2=0, \quad a'x^2+2h'xy+b'y^2=0 \] may represent a harmonic pencil. \par Shew that the same condition would make the first pair of lines conjugate diameters of the conic \[ a'x^2+2h'xy+b'y^2=1. \] Explain the relation of these results in reference to the theory of pencils in involution.
If a circle \(S\) touches the circumscribed circle of a triangle \(ABC\) at \(P\), prove that the tangents to \(S\) from \(A, B, C\) are in the ratio \(AP:BP:CP\). \par What does the result become when the radius of the circle \(S\) tends to infinity?
Prove that if in a plane the ratio of the distances from two points are the same for each of the three points \(A, B, C\), the two points are inverse points with regard to the circle \(ABC\). \par Shew that the line bisecting \(BC\) at right angles meets the lines \(BA\) and \(CA\) in two inverse points.
Prove that the locus of the feet of perpendiculars from a focus to tangents to a conic is a circle. \par The tangent to this circle at \(Y\) meets the major axis of the conic in \(R\), and tangents to the conic from \(Y\) touch at \(P\) and \(Q\); shew that \(P, Q, R\) are collinear.
Shew that the reciprocal of a circle with respect to a circle is a conic. \par Reciprocate the following theorem with respect to any point:-- \par Two circles cut at \(A, B\), and a line touches them at \(C, D\). \(CA\) cuts one circle in \(E\) and \(DB\) cuts the other in \(F\). Then \(CF\) and \(DE\) are parallel.
Prove the constant cross ratio property of four points of a conic. \par \(ABC\) is a triangle and \(P\) any point on the circumcircle. \(BP\) meets \(AC\) in \(M\) and \(CP\) meets \(AB\) in \(N\). Shew that \(MN\) envelops a conic.
The equation of two lines is \(ax^2+2hxy+by^2=0\); find the equation of the lines bisecting the angle between them. \par If \(lx+my=1\) bisects the angle between two lines one of which is \(px+qy=1\), shew that the other line is \[ (px+qy-1)(l^2+m^2) = 2(pl+qm)(lx+my-1). \]
Find the radical axis of the circles \[ x^2+y^2-4x-2y+4=0, \quad x^2+y^2+4x+2y-4=0, \] and the coordinates of the limiting points of the coaxal system to which they belong. \par Find the equation of the circle which touches them externally and passes through the point (1, 2).
Shew that the normal to a parabola at the point \(x=am^2, y=2am\) is \(y+mx=2am+am^3\). \par The tangent at \(P\) to the above parabola makes an angle \(\psi\) with the axis of the curve. The normal at \(P\) meets the curve at \(Q\). Shew that \(PQ\) is equal to \(4a/\sin\psi\cos^2\psi\) and that the least value of \(PQ\) is \(6a\sqrt{3}\).