Prove the harmonic property of pole and polar with respect to a circle. Having given a point and its polar, and a point on the circle, construct the circle.
Prove that if two circles cut orthogonally, any line through the centre of either is divided harmonically by the other. Also prove conversely that if a line is cut harmonically by two orthogonal circles it must pass through the centre of one of them.
Prove that the polar reciprocal of one circle with respect to another is a conic, and shew how to find its centre and asymptotes. A rectangular hyperbola circumscribes a triangle. With any point \(P\) on the hyperbola as focus a conic \(S\) is described so as to touch the sides of the triangle. If this figure is reciprocated from \(P\), state what the reciprocal figure becomes, and prove that the auxiliary circle of the conic \(S\) passes through the centre of the rectangular hyperbola.
Prove that the straight lines joining the middle points of the opposite edges of a tetrahedron meet in a point and bisect one another. Also prove that if these three straight lines are equal, each edge of the tetrahedron is orthogonal to the opposite edge.
Find the condition that the pair of straight lines represented by the equation \[ ax^2+2hxy+by^2=0 \] should be perpendicular. A chord of a circle subtends a right angle at a given eccentric point \(O\); prove that the locus of the foot of the perpendicular drawn to the chord from \(O\) is a circle.
Chords are drawn from the origin to the parabola \(2y = ax^2+2bx+c\); prove that their middle points lie on \(y=ax^2+bx\).
Find the equation of the straight line joining two points on an ellipse whose eccentric angles are given. \(OP, OQ\) are conjugate diameters of an ellipse; \(R\) is a point on the ellipse the tangent at which is parallel to \(PQ\). Shew that the locus of the centre of gravity of the triangle \(PQR\) is a similar ellipse.
Prove that the coordinates of the foci of the conic \(ax^2+2hxy+by^2=1\) may be found from the equations \[ \frac{x^2-y^2}{a-b} = \frac{xy}{h} = \frac{1}{h^2-ab}. \]
If \(S=0\) is a conic, \(T=0\) a tangent to the conic and \(\alpha=0\) a straight line, interpret the equations \(S-kT=0, S-k\alpha T=0, S-kT^2=0\), where \(k\) is a constant. Find the locus of the centres of conics which pass through four given points.
Prove that the three feet of the perpendiculars on the sides of a triangle from any point on its circum-circle lie on a straight line (the pedal line). Prove that, if \(P\) be any point on the circum-circle of the triangle \(ABC\), and \(PQ\) be drawn parallel to \(BC\) to meet the circum-circle in \(Q\), then will \(QA\) be perpendicular to the pedal line of \(P\) with respect to the triangle.