In connection with the method of inversion prove that: (i) the inverse of a circle is a circle or a straight line; (ii) the angle of intersection of two circles is unaltered by inversion. Shew that the problem of drawing a circle to pass through a given point and to touch two given circles admits of four solutions.
Give examples to illustrate the utility of the method of reciprocation in geometry.
Shew that the general equation of the first degree in Cartesian coordinates represents a straight line and also that the equation of any straight line is of the first degree. \(A, B, C, D\) are four fixed points and \(P\) is another point such that the sum of the areas of the triangles \(PAB\) and \(PCD\) is constant, then the locus of \(P\) is a certain portion of a straight line. What property has \(P\) in relation to the triangles when it lies on the other parts of the straight line?
Shew that the equation of the circle on the line joining \((x_1, y_1)\) to \((x_2, y_2)\) as diameter is \((x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0\). Shew that if \(P\) and \(Q\) are conjugate points with respect to a circle \(S\), then the circle on \(PQ\) as diameter cuts \(S\) at right angles.
Shew that an equation of the form \(y^2+2ax+2by+c=0\), represents a parabola. Prove that if the axes of two parabolas are at right angles, then their four common points lie on a circle.
Shew that four normals can be drawn from any point \(O\) to an ellipse and that their feet lie on a rectangular hyperbola which passes through \(O\) and the centre and has its asymptotes parallel to the axes of the ellipse. Shew also that the locus of the middle points of the chords of the ellipse whose ends are equally distant from \(O\) is the same hyperbola.
Give an account of a method by which it is proved that the general equation of the second degree is the equation of a conic. Shew that by a proper choice of rectangular axes the general equation of the second degree can usually be reduced to the form \(y' = \alpha x'^2 + \beta x'\). Indicate the geometrical significance of \(\alpha\) and \(\beta\) and point out any exceptional cases.
Shew that if the opposite edges of a tetrahedron are at right angles then the perpendiculars from the vertices meet in a point.
Give a geometrical construction for finding two lengths, having given their sum and the mean proportional between them. \(ABC\) is a triangle, right-angled at \(A\); \(BDEC\) is a square on the remote side of \(BC\); if \(AD, AE\) cut \(BC\) in \(P, Q\) respectively, shew that \(PQ\) is a mean proportional between \(BP\) and \(QC\).
Shew that a chord of a circle through any point is harmonically divided by the point, its polar, and the circle. \(ABC\) is a triangle inscribed in a circle and \(P\) is the pole of \(BC\). Shew that the chord through \(P\) parallel to \(AB\) is bisected by \(AC\).