Define the polar of a point with respect to a circle. Prove that any line cutting a circle and passing through a fixed point is cut harmonically by the circle, the point and the polar of the point. If \(A, B\) are given points on a circle, and if \(CD\) is a given diameter, shew how to find a point \(P\) on the circle such that \(PA, PB\) shall cut \(CD\) in points equidistant from the centre.
Prove that the inverse of a circle is a straight line or a circle. \(S'\) is the inverse of a circle \(S\) with respect to a circle centre \(O\). \(T\) and \(T'\) are the lengths of the tangents from \(P\) and \(P'\) to \(S\) and \(S'\) respectively where \(P'\) is the inverse of \(P\). Prove that \(\frac{T}{T'} = \frac{t'}{OP}\), where \(t'\) is the length of the tangent from \(O\) to \(S'\).
Prove that the curve of intersection of two spheres is a circle. If three circles in space are such that each intersects each of the others in two points, prove that the three circles all lie on the same sphere.
Prove that the sum of the squares of a pair of conjugate diameters of an ellipse is constant. A rectangle circumscribes an ellipse. Show that the rectangle contained by the segments into which a side is divided by the point of contact is equal to the square of the semi-diameter of the ellipse parallel to that side.
Find the length of the perpendicular from the point \((h,k)\) to the line \(u=ax+by+c=0\). If \(u'=a'x+b'y+c'=0\), find the equation of the line which makes the same angle with \(u'=0\) that \(u-\lambda u'=0\) makes with \(u=0\).
Find the equation of the normal to the parabola \(y^2=4ax\) at the point \((at^2, 2at)\). The normals at the points \(P, Q\) on the parabola intersect on the parabola. Prove that the locus of the middle point of \(PQ\) is \(y^2=2a(x+2a)\).
Prove that the chords of intersection of a circle and a conic are equally inclined in pairs to the axes of the conic. Find the equation of the circle which passes through the origin and touches the ellipse \(x^2/a^2+y^2/b^2=1\) at an extremity of a latus rectum and show that it cuts the major axis again at a distance \(ae(1+e^2-e^4)\) from the centre.
Find the condition that \(ax^2+2hxy+by^2+2gx+2fy+c=0\) should represent an ellipse, parabola or hyperbola. When it represents a parabola find the equation of its axis and the coordinates of its focus.
Prove that the angles made by a tangent to a circle with a chord drawn from the point of contact are respectively equal to the angles in the alternate segments of the circle. Two circles touch at \(A\), one being inside the other; the tangent at a point \(B\) to the inner circle cuts the outer circle in \(C\) and \(D\). Prove that the angles \(BAC\) and \(BAD\) are equal.
Prove that, if the polar of a point \(P\) with respect to a circle passes through the point \(Q\), the polar of \(Q\) will pass through the point \(P\). Prove also that the circle described on \(PQ\) will cut the given circle at right angles.