Problems

Two conics intersect in four points \(A, B, C, D\). Shew that if the tangents at \(A, B\) to the first conic meet on the second conic, so do the tangents to the first conic at \(C, D\).

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 \(ax^2+by^2+2hxy+2gx+2fy+c=0\):

1. [(i)] The principal axes are given by the equation \[ \frac{\left(x-\frac{G}{C}\right)^2-\left(y-\frac{F}{C}\right)^2}{a-b} = \frac{\left(x-\frac{G}{C}\right)\left(y-\frac{F}{C}\right)}{h}, \] where \(C=ab-h^2, G=hf-bg, F=hg-af\).
2. [(ii)] If \(C\) is positive, the area enclosed by the curve is \(\frac{\pi D}{C^{3/2}}\), where \[ D = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}. \]

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:

1. [(a)] \(S_1 = S_2\);
2. [(b)] \(\lambda_1 S_1 + \lambda_2 S_2 = 0\);
3. [(c)] \(S_1=S_2=S_3\);
4. [(d)] \(\lambda S_1 + \mu S_2 + \nu S_3 = 0\).

Shew that in general it is possible to find one circle orthogonal to three given circles, and investigate any exceptional cases. \par Determine the equation of the circle which is orthogonal to the three circles \begin{align*} x^2+y^2 &= 2x; \\ x^2+y^2+8x-12y+36 &= 0; \\ x^2+y^2+4x+2y-6 &= 0. \end{align*}

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 \(2s=a+b+c+d\).