If \(a_1, \ldots, a_n\) and \(b_1, \ldots, b_n\) are real numbers prove, by considering the minimum value as \(x\) varies of \(\sum_{r=1}^n (xa_r + b_r)^2\), or otherwise, that $$\left(\sum_{r=1}^n a_r^2\right)\left(\sum_{r=1}^n b_r^2\right) > \left(\sum_{r=1}^n a_r b_r\right)^2.$$ Hence prove by analytical geometry that, if \(ABC\) is a triangle, \(AB + BC > AC\). Two circles, \(C_1\) and \(C_2\), touch at \(T\). A variable circle \(C\) goes through \(T\) and cuts \(C_1\) and \(C_2\) again orthogonally in \(X\) and \(Y\). Prove that in general \(XY\) passes through a fixed point. Also discuss the exceptional case.
A variable chord \(AB\) of a conic subtends a right angle at a fixed point \(O\). Show that in general the foot of the perpendicular from \(O\) to \(PQ\) lies on a circle. What is the exceptional case?
\(ABCD\) is a tetrahedron. \(O\) is a point not lying on any of its faces. The line through \(O\) and \(A\) cuts \(BCD\) in \(P\) and \(P'\), respectively. Similarly the line through \(O\) meeting \(BC\) and \(AD\) cuts them in \(Q\) and \(Q'\) respectively, and that meeting \(AB\) and \(CD\) cuts them in \(R\) and \(R'\), respectively. Prove that \(AP\), \(BQ\), \(CR\) and \(DO\) are concurrent.
\(T\) is a point on a parabola of which \(S\) is the focus. A circle through \(S\) and \(T\) cuts the tangent to the parabola at \(T\) again in \(U\). Prove that the tangent to the circle at \(U\) is also a tangent to the parabola.
tangent to the parabola.
Given an ellipse, describe how to find its centre, axes and foci using ruler and compasses only, and justify your constructions.
Given a triangle \(ABC\), points \(Q\), \(M\) are taken on the side \(AC\) such that \(AQ = \frac{1}{4}AC\), \(AM = \frac{2}{3}AC\), and points \(R\), \(N\) are taken on the side \(AB\) such that \(AR = \frac{1}{4}AB\), \(AN = \frac{3}{4}AB\). The line \(BQ\) meets \(CR\), \(CN\) in \(X\), \(Y\) respectively; the line \(BM\) meets \(CR\), \(CN\) in \(U\), \(T\) respectively. Prove that \(XY\) bisects \(BC\) and that \(UV\) is parallel to \(BC\).
Three points \(A\), \(B\), \(C\) are given in general position in a plane. A circle of the coaxal system with \(A\), \(B\) as limiting points meets a circle of the coaxal system with \(A\), \(C\) as limiting points in \(U\), \(V\). Prove that the line \(UV\) passes through the circumcentre of the triangle \(ABC\).
Three concurrent lines \(DA\), \(DB\), \(DC\) in space are such that each is perpendicular to the other two. Identify the common chord of the three spheres with diameters \(BC'\), \(CA\), \(AB\).
The ellipse \(x^2/a^2 + y^2/b^2 = 1\) has foci \(S(ae, 0)\) and \(S'(-ae, 0)\); \(P(x_1, y_1)\), where \(x < ae\), is an arbitrary point of the ellipse (taken, for convenience, in the first quadrant) and \(Q\) is the point \((x_1, -y_1)\). Prove that a circle can be drawn to touch each of the segments \(SP\), \(S'P\), \(SQ\), \(S'Q\). Find the centre of the circle, and prove that its radius is \(ey_1\).