Shew that if points in a straight line \(OX\) are connected in pairs \((P,Q)\) by the one-one relation \(axy+b(x+y)+c=0\), where \(x=OP, y=OQ\), then the cross ratio of four points \((P_1P_2P_3P_4)\) is equal to the cross ratio of the four corresponding points \((Q_1Q_2Q_3Q_4)\). \par Derive the existence of double points \((L, M)\) which are harmonic conjugates with respect to any pair \((P,Q)\).
Shew that for the conic given by the equation \(ax^2+by^2+2hxy+2gx+2fy+c=0\):
State the condition that the equation \(ax^2+by^2+2hxy+2gx+2fy+c=0\) shall represent two straight lines, and prove that it is both necessary and sufficient. \par The equation of the straight lines \(AB, AD\) is \(bx^2+ay^2=0\), and that of the straight lines \(CB, CD\) is \(ax^2+by^2+2hxy+2gx+2fy+c=0\). Find the equation of the two diagonals of the quadrilateral \(ABCD\) which do not pass through \(A\), and deduce that they will be perpendicular if \((a+b)(f^2+g^2)=2fgh+c(a^2+b^2-h^2)\).
If \(S_r \equiv x^2+y^2+2g_rx+2f_ry+c_r\), interpret geometrically the following equations:
Define a parabola and deduce the parametric representation in the usual form \((at^2, 2at)\). \par A circle is drawn through the vertex of a parabola and cuts the curve in three other points \(P, Q, R\). Shew that the normals to the parabola at \(P, Q, R\) are concurrent. \par Shew also that if a circle is drawn through any two fixed points on a parabola, the midpoint of the join of the other pair of intersections lies on a fixed straight line.
Shew that of the family of confocal conics given by the equation \(\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1\), two and only two members pass through a given point, one a hyperbola and one an ellipse. Derive the expression for the coordinates \((x,y)\) of the point in terms of the parameters \(\lambda_1, \lambda_2\) corresponding to the two conics in the form: \[ x^2 = \frac{(a^2+\lambda_1)(a^2+\lambda_2)}{a^2-b^2}, \quad y^2 = \frac{(b^2+\lambda_1)(b^2+\lambda_2)}{b^2-a^2}. \] Shew that the semi-axis of the ellipse parallel to the normal at \((x,y)\) to the hyperbola is such that the square of its length is \(\lambda_1 \sim \lambda_2\).
Shew that \(x^2-2x\cos\theta+1\) is a factor of \(x^{2n}-2x^n\cos n\theta+1\), and find the other real quadratic factors of this expression. \par Hence, or otherwise, obtain the results: \begin{align*} \sin n\theta &= 2^{n-1} \prod_{s=0}^{n-1} \left[\sin\left(\theta+\frac{s\pi}{n}\right)\right]; \\ \cos n\theta &= 2^{n-1} \prod_{s=0}^{n-1} \left[\sin\left(\theta+\frac{(2s+1)\pi}{2n}\right)\right]. \end{align*}
Shew that a quadrilateral with sides of given lengths has its greatest area when it is cyclic. \par Shew further that the area of a cyclic quadrilateral is \(\sqrt{(s-a)(s-b)(s-c)(s-d)}\), where \(a,b,c,d\) are the lengths of the sides, and \(2s=a+b+c+d\).
Two variable points \(P, Q\) on a fixed line subtend a constant angle at a fixed point \(O\); prove that the variable circle \(OPQ\) touches a fixed circle, with respect to which \(O\) and the reflexion of \(O\) in the fixed line are inverse points.
``A tangent to a circle is perpendicular to the radius through its point of contact'': reciprocate this property with respect to any other circle. \par A variable tangent \(t\) to a conic meets a fixed tangent at \(P\); find the locus of intersection of \(t\) and the line through a focus \(S\) perpendicular to \(SP\).