Prove that the inverse of a sphere with respect to any internal point is a sphere. Invert with respect to the orthocentre the theorem that the three perpendiculars of a triangle are concurrent.
Shew that the cross-ratio of the pencil \(u+\lambda_r v=0\), (\(r=1,2,3,4\)), is \[ \frac{(\lambda_1-\lambda_2)(\lambda_3-\lambda_4)}{(\lambda_1-\lambda_4)(\lambda_3-\lambda_2)}. \] The equations of the sides \(AB, AC\) of a triangle \(ABC\) are \(u=0\) and \(v=0\) respectively, and the equation of the internal bisector of the angle \(A\) is \(u+v=0\). Shew that the equation of the straight line through \(A\) and perpendicular to \(BC\) is \(u\cos C+v\cos B=0\). Obtain an expression for the cross-ratio of the pencil formed by these four lines.
\(2a\) and \(2b\) are respectively the lengths of the major and minor axes of an ellipse which touches a parabola of latus rectum \(4c\). When produced the major axis of the ellipse coincides with the axis of the parabola. Shew that the length intercepted by the points of contact on a common tangent to the two curves (other than the tangent at their point of contact) is \[ \frac{(2ac+b^2)^{\frac{1}{2}}(2ac+b^2+c^2)^{\frac{1}{2}}}{c(ac+b^2)^{\frac{1}{2}}}. \]
The straight line passing through the points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) intersects the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) in the points \(P\) and \(Q\). Obtain an equation whose roots are the ratios in which \(AB\) is divided by \(P\) and \(Q\). Hence find the equation of the pair of tangents drawn from \(A\) to the ellipse and deduce the equation of the director circle. Shew that the envelope of the chords of contact of tangents drawn to an ellipse of eccentricity \(e\) from points on its director circle is an ellipse whose eccentricity is \(e(2-e^2)^{-\frac{1}{2}}\).
The line \(lx+my+n=0\) intersects the circle \(x^2+y^2+2gx+2fy+c=0\) in \(A\) and \(B\). \(O\) is the origin, and the straight lines \(OA, OB\) intersect the circle again in \(P\) and \(Q\). Shew that the equation of the straight line \(PQ\) is \[ c(lx+my+n)=2n(gx+fy+c). \] As \(n\) varies find the least distance of the point of intersection of \(AB\) and \(PQ\) from the origin, stating the corresponding value of \(n\).
\(ax+by+c=0\) is one asymptote of a hyperbola which passes through the origin and which touches the straight line \(bx+ay=c\) at the point \(\left(\dfrac{c}{2b}, \dfrac{c}{2a}\right)\). Find the equation of the hyperbola and of the normal at the origin. Prove that if this normal is parallel to the other asymptote then \((a-b)^4 = 8a^2b^2\).
Shew that if the general equation of the second degree represents a parabola then the terms of the second degree form a perfect square. Find the equation of the reflection of the parabola whose equation is \[ 16x^2+9y^2+24xy-81x+18y+34=0 \] in the tangent at its vertex.
\(\Delta\) and \(R\) are respectively the area and the circumradius of a triangle \(ABC\); \(\delta\) and \(r\) are respectively the area and the inradius of the pedal triangle of \(ABC\). Prove that \(\dfrac{\delta}{\Delta} = \dfrac{r}{R} = x\), where \(x\) is a root of the equation \[ x^2+4x\cos^2 B\cos^2 C + 2\cos^2 B\cos^2 C(\cos 2B+\cos 2C) = 0. \] Write down the other root of this equation in terms of \(B\) and \(C\).
Obtain the \(n\)th roots of \(a+b\sqrt{-1}\), where \(a\) and \(b\) are real. If \(\omega\) is one of the imaginary \(2n\)th roots of unity, prove that \[ \sin\theta+\omega\sin 2\theta+\omega^2\sin 3\theta+\dots+\omega^{2n-1}\sin 2n\theta = \frac{\sin\theta+\omega\sin 2n\theta - \sin(2n+1)\theta}{1-2\omega\cos\theta+\omega^2}. \] Deduce the sums of the series \[ \sin\theta+\omega^2\sin 3\theta+\omega^4\sin 5\theta+\dots+\omega^{2n-2}\sin(2n-1)\theta \] and \[ \sin 2\theta+\omega^2\sin 4\theta+\omega^4\sin 6\theta+\dots+\omega^{2n-2}\sin 2n\theta, \] and find all the values of \(\theta\) which make these two sums equal.