Find the directions and magnitudes of the principal axes of the conic \(ax^2+2hxy+by^2=1\). Find also the angle between the equiconjugate diameters.
Interpret the equations \(S-LM=0, S-L^2=0\); where \(S=0\) is a conic, and \(L=0, M=0\) are straight lines. Find the equation of the circle of curvature at the point on \(x^2/a^2+y^2/b^2=1\) whose eccentric angle is \(\phi\). Find also the eccentric angle of the other point in which the circle cuts the ellipse.
Show that the locus of a point such that the lengths of the tangents from it to two circles are equal, is a straight line. \(AB, CD\) are diameters of two circles and \(AC\) is parallel to \(BD\). Prove that \(AD, BC\) meet on the radical axis of the circles.
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 \(\frac{x^2}{a^2}+\frac{2y^2}{b^2}=1\).