A is a point on a given circle. Shew how, with ruler and compasses, to find another point P on the circle such that the sum of the distances of P from A and from the tangent at A shall be equal to a given length not greater than twice the diameter of the circle.
A circle is drawn to cut the auxiliary circle of an ellipse at right angles and to touch the ellipse at P. The normal PG to the ellipse at P cuts the circle again in Q. Prove that GQ is equal to the radius of curvature of the ellipse at P.
Prove that if P be any point of a hyperbola whose foci are S and H, and if the tangent at P meets an asymptote in T, the angle between that asymptote and HP is twice the angle STP.
An ellipse inscribed in an acute-angled triangle ABC has one focus at the orthocentre. Prove that the square of the semidiameter of the ellipse parallel to the side BC is \[ 2R^2 \cos A \cos B \cos C \sec^2(B-C), \] where R is the radius of the circumcircle of the triangle.
PT, PT' are the tangents to an ellipse from a point P on one of the equiconjugate diameters. Prove that the circle PTT' passes through the centre.
Prove that if \(x^4 + ax^2 + bx + c\) is divisible by \(x^2+px+q\) and a, b, p are given then q and c are uniquely determinate in general. Carefully examine the exceptional cases.
Prove that, if x and y are unequal, and \[ x(1-yz) = (x^2-1)(y+z), \quad y(1-zx)=(y^2-1)(z+x), \] then \[ z(1-xy)=(z^2-1)(x+y). \]
Shew that if \[ y^2+yz+z^2 = a^2, \quad z^2+zx+x^2=b^2, \quad x^2+xy+y^2=c^2, \] then \(x+y+z\) is given by the equation \[ (x+y+z)^4 - (x+y+z)^2(a^2+b^2+c^2) + a^4+b^4+c^4 - b^2c^2 - c^2a^2 - a^2b^2 = 0, \] and hence indicate how to solve the above equations.
A set of numbers \(a_1, a_2, a_3, \dots, a_n, \dots\), is such that from the third onwards each is the arithmetic mean between its two immediate predecessors: prove that \[ a_n = A+B(-\frac{1}{2})^n, \] where A and B are independent of n and are to be found in terms of \(a_1\) and \(a_2\).
Prove that, if \begin{align*} a \cos x \cos y + b \sin x \sin y &= c, \\ a \cos y \cos z + b \sin y \sin z &= c, \\ a \cos z \cos x + b \sin z \sin x &= c, \end{align*} where \(a, b, c\) are not all equal, then \(bc+ca+ab=0\).