Prove that the polar reciprocal of a circle is a conic of which the origin of reciprocation is a focus. Prove that two ellipses with one common focus cannot intersect in four real points.
Find the equation of the normal, the coordinates of the centre of curvature and the equation of the circle of curvature at the point \((am^2, 2am)\) on the parabola \(y^2-4ax=0\). Shew that this circle of curvature surrounds the circle of curvature at the vertex and their radical axis touches the parabola \(2y^2=9ax\).
Prove that there are two points on a quadrant of an ellipse such that the normals are at the same given distance from the centre and that the distance must be less than \(a-b\); shew also that if \(\theta_1, \theta_2\) be the excentric angles of the feet of such a pair of normals, \(a\tan\theta_1\tan\theta_2=b\), where \(2a, 2b\) are the lengths of the axes.
Prove that the ellipse \[ b^2x^2+a^2y^2=a^2b^2, \quad b^2 = a^2(1-e^2) \] is touched at two points by each of the circles \[ x^2+y^2-2\lambda aex + \lambda^2a^2 = b^2, \] \[ x^2+y^2-2\mu aey - \mu^2b^2 = a^2, \] and that, if one point of contact be common to the two circles, \[ a^2\lambda^2+b^2\mu^2 = a^2-b^2. \]
Prove that the equation of any conic inscribed in the rectangle \[ x = \pm a, \quad y = \pm b \] is of the form \[ b^2x^2+a^2y^2-a^2b^2-2\lambda xy + \lambda^2=0: \] and that the two conics of the system, defined by \(\lambda_1\) and \(\lambda_2\), have their eight points of contact with the sides of the rectangle on the conic \[ b^2x^2+a^2y^2-a^2b^2 - (\lambda_1+\lambda_2)xy+\lambda_1\lambda_2=0. \]
A family of conics is such that two given points are the respective poles of two given lines with regard to each conic: shew that the conics have double contact, touching two definite lines at the same points, and that this relation passes into four-point contact if the line joining the two given poles passes through the intersection of the polars.
Of two circles which cut orthogonally one has a fixed centre and the other passes through two fixed points. Shew that their radical axis passes through a fixed point, and determine its position.
Given an obtuse-angled triangle, determine a circle of which it is the self-conjugate triangle. Show that the circle is real and that there cannot be more than one.
Find the sum of the cubes of the first \(n\) natural numbers, and determine a set of \(2n+1\) consecutive integers such that their sum bears to the sum of their cubes the ratio \(1:m^2+n^2+n\).
Prove that in any triangle \(ABC\), \[ \cos A + \cos B + \cos C \le \frac{3}{2}, \] \[ \cot B \cot C + \cot C \cot A + \cot A \cot B \ge 9. \]