Tangent lines are drawn to a sphere from an external point. Prove that the points of contact lie on a circle. Prove that, if \(PA, PB, PC\) be three lines touching a sphere at \(A, B, C\) and mutually at right angles, then \[ PA^2 + PB^2 + PC^2 = \frac{3}{4}(\text{diameter})^2. \]
Prove that the cross ratio of the pencil formed by joining a variable point on a conic to four fixed points on the conic is constant. \(ABC\) is a fixed triangle inscribed in a conic. \(O\) is a fixed point on the conic and any line through \(O\) cuts the conic in \(P\) and the sides of the triangle in \(X, Y, Z\). Prove that the cross ratio \((PXYZ)\) is constant.
Prove that the polar reciprocal of a conic with regard to the focus is a circle. \(ABC\) is a triangle circumscribing an ellipse. \(AB\) touches the curve at \(F\). \(S\) is a focus. Prove that the angles \(ASF, BSC\) are supplementary.
Interpret the expression \(x^2+y^2+2gx+2fy+c\) in which \(x, y\) are coordinates of any point in a plane. Prove that the locus of a point such that the tangents from it to two given circles are in a constant ratio is a circle coaxal with the given circles.
Two normals to a parabola make angles \(\theta, \theta'\) with the axis. Prove that, if \(\tan\theta\tan\theta'=2\), the normals meet on the curve. \(TP, TQ\) are tangents to the parabola \(y^2=4ax\) and the normals at \(P, Q\) meet on the curve in \(R\). Prove that the locus of the middle point of \(TR\) is the parabola \(2y^2=a(x-a)\).
Find the equation of the tangent at any point on the ellipse \(x=a\cos\phi, y=b\sin\phi\). The tangent at \(P\) meets the directrices in \(E\) and \(F\). Prove that the other tangents from \(E\) and \(F\) intersect on the normal at \(P\).
Prove that if the equation of a conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] be transformed by any change of coordinate axes, the following expressions remain unaltered in value, \[ (a+b-2h\cos\omega)\text{cosec}^2\omega, \quad (ab-h^2)\text{cosec}^2\omega, \] and \((abc+2fgh-af^2-bg^2-ch^2)\text{cosec}^2\omega\); where \(\omega\) is the angle between the original axes. Hence, or otherwise, shew that, if the given equation represent a parabola, the length of its latus rectum is \[ \frac{2(\sqrt{af}\sim\sqrt{bg})\sin^2\omega}{(a+b-2h\cos\omega)^{\frac{3}{2}}}. \]
Find the locus of the polar of a given straight line with regard to a system of confocal conics. Tangents are drawn to an ellipse from a point \(P\). Prove that the bisectors of the angles between the tangents are normals to the conics through \(P\) confocal with the ellipse.
\(O\) is the circumcentre, \(G\) the centroid and \(H\) the orthocentre of a triangle. Prove that \(O, G\) and \(H\) are collinear, and that \(HG=2GO\). Prove also that the nine-points centre is collinear with the other points, and that it bisects \(OH\).
Prove that the volume of a parallelepiped constructed by drawing through the opposite edges of a tetrahedron three pairs of parallel planes, is three times the volume of the tetrahedron. In a tetrahedron \(ABCD\), if \(AB\) is perpendicular to \(CD\), and \(AC\) is perpendicular to \(BD\), prove that \(AD\) is perpendicular to \(BC\).