In a tetrahedron show that the perpendicular to any face through its orthocentre intersects all the perpendiculars from the vertices on the opposite faces.
Show that the inverse of a circle with respect to a point is a straight line or a circle. Show also that a pair of points inverse with respect to the circle invert into a pair of points inverse with respect to the inverse circle. If \(A', B'\) are the inverse points of \(A, B\) with respect to a circle and \(P\) is any point upon it, show that the circles \(PAB, PA'B'\) meet again upon the circle.
A circle cuts an ellipse in four points. Prove that the line joining two of the points and the line joining the other points make equal angles with either axis. Given an ellipse, give a geometrical construction for its axes.
Prove that if \(A, B, C, D\) are four fixed points on a conic, and \(P\) is any point of the curve, the cross ratio of the pencil \(P\{ABCD\}\) is the same for all positions of \(P\). Show that if conics are drawn through four points \(A, B, C, D\) and the tangents at \(A\) and \(B\) to any such conic meet in \(P\), the locus of \(P\) is a straight line.
Show that the line \((x-a)\cos\phi+y\sin\phi=b\) touches the circle \((x-a)^2+y^2=b^2\). A pair of parallel tangents is drawn to a circle and another pair of parallel tangents perpendicular to the first pair is drawn to another equal circle. Prove that each of the diagonals of the square formed by the four tangents passes through a fixed point.
Find the equations of the tangent and normal to the parabola \(y^2=4ax\) at the point \((at^2, 2at)\). From the point \(P(at^2, 2at)\) of this parabola two chords \(PQ, PR\) are drawn normal to the curve at \(Q, R\). Prove that the equation of \(QR\) is \(yt+2(x+2a)=0\).
The lines \(lx+my=1\) and \(l'x+m'y=1\) are such that each passes through the pole of the other with respect to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). Show that \(a^2ll'+b^2mm'=1\). If the two lines above pass respectively through the ends of the major axis of the ellipse, show that they intersect on the ellipse \(\frac{x^2}{a^2}+\frac{2y^2}{b^2}=1\).
Find the lengths of the axes of the conic \(ax^2+2hxy+by^2=1\). An ellipse of semi-axes \(u,v\) revolves round its centre in its own plane. Show that the locus of the poles of a fixed straight line whose distance from the centre is \(c\) is a circle of radius \(\frac{u^2-v^2}{2c}\).
The equations of the sides of a triangle are \(\alpha=0, \beta=0, \gamma=0\). Show that the equation of any conic which touches the three sides is \(\sqrt{l\alpha}+\sqrt{m\beta}+\sqrt{n\gamma}=0\). Show that two triangles whose sides pass respectively through \(A, B, C\) can be inscribed in the above conic and that the equations of the lines joining corresponding angular points of the triangles are \(l\alpha+m\beta-3n\gamma=0\), and two similar equations.
Complex Numbers.