A sphere of radius \(R\) rolls between two fixed horizontal straight lines which intersect at an angle \(2\alpha\). Prove that, until the sphere slips through, its centre describes an ellipse of minor axis \(R\) and eccentricity equal to \(\cos\alpha\).
Find the angle between the two straight lines \[ ax^2+2hxy+by^2=0. \] Prove that the equation of the straight lines through the origin that make equal angles \(\alpha\) with the line \(x+y=0\) is \[ x^2+2xy\sec 2\alpha+y^2=0. \]
Prove that the equation of a family of coaxal circles can be expressed in the form \[ x^2+y^2+2\mu x+c=0, \] where \(\mu\) is variable; distinguish between the cases when \(c\) is positive, negative or zero. Prove that the locus of the poles of a given straight line with respect to the circles of the family is in general a hyperbola, with asymptotes respectively parallel to the radical axis and perpendicular to the given straight line.
Find the equation of the chord of the parabola \(y^2=4ax\) which is bisected at the point \((x', y')\). Through every point of the straight line \(x=my+h\) is drawn a chord of the parabola \(y^2=4ax\) which is bisected at the point; prove that these chords touch the parabola \[ (y+2am)^2=8a(x-h). \]
Find the equation of the pair of tangents that can be drawn from \((x',y')\) to the conic \(px^2+qy^2=1\). If the pair of tangents is always parallel to conjugate diameters of the conic \[ ax^2+2hxy+by^2=1, \] prove that the locus of \((x',y')\) is \[ ax^2+2hxy+by^2 = \frac{a}{p}+\frac{b}{q}. \]
A chord of a hyperbola subtends a right angle at a fixed point \(O\) not on the curve. Prove that (i) the locus of the foot of the perpendicular from \(O\) on the chord is a circle, and (ii) a pair of common chords of the circle and the hyperbola meet in \(O\), and are at right angles to the asymptotes of the hyperbola. Examine the case when \(O\) is on the hyperbola.
Four points \(A,B,A',B'\) are given in a plane: prove that there are always two positions of a point \(C\) in the plane such that the triangles \(CAB, CA'B'\) are similar, the equal angles being denoted by corresponding letters.
Show that chords of a circle through a fixed point are cut harmonically by the point, its polar, and the circle. \(ABC\) is a triangle inscribed in a circle and \(O\) is the pole of \(AB\). Show that the chord through \(O\) parallel to \(AC\) is bisected by \(BC\).
Prove that the locus of the foot of the perpendicular from a focus of an ellipse on any tangent is a circle on the major axis as diameter (auxiliary circle). Given a focus of a conic, two tangents, and a point on the auxiliary circle, give a geometrical construction for the other focus. Show also how to construct any number of points on the conic.
Prove that the reciprocal of a circle is a conic with a focus at the centre of reciprocation. Find the position of the centre of reciprocation if the conic is a rectangular hyperbola. Show that a chord of a rectangular hyperbola which subtends a right angle at a focus touches a fixed parabola.