\(S\) is the focus of a parabola, and the normal at \(P\) meets the axis in \(G\). Prove that \(\frac{SG}{SP}\) is equal to the eccentricity, and that \(\frac{PG^2}{SP}\) is equal to the latus rectum.
\(P\) is any point on an ellipse whose major axis is \(AA'\), and whose foci are \(S\) and \(S'\). Prove that the centres of two of the escribed circles of the triangle \(SPS'\) lie on the tangents to the ellipse at \(A\) and \(A'\); and that the locus of the centre of the inscribed circle of the triangle \(SPS'\) is an ellipse.
Prove that the two straight lines \(x^2(\tan^2\theta+\cos^2\theta)-2xy\tan\theta+y^2\sin^2\theta=0\) form with the line \(x=c\) a triangle of area \(c^2\).
Find the equation of the two tangents that can be drawn from \((x',y')\) to the parabola \(y^2=4ax\). On the axis \(OX\), and outside the parabola, any two points \(P\) and \(Q\) are taken such that the rectangle \(OP.OQ\) is constant and equal to \(c^2\). Prove that the locus of the four points of intersection, other than \(P\) and \(Q\), of the four tangents from \(P\) and \(Q\) to the parabola is the pair of straight lines \(x^2=c^2\).
Find the equation of the normal to the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \] at the point \(P(x', y')\). If the tangents at \(P(x',y')\) and \(Q(x'',y'')\) meet at \((X,Y)\), prove that the normals at \(P\) and \(Q\) meet at \[ \left( \frac{(a^2-b^2)x'x''X}{a^4}, \frac{(b^2-a^2)y'y''Y}{b^4} \right). \]
Prove that the equation \[ \frac{l}{r} = 1+e\cos\theta \] represents in polar coordinates a conic with the origin as focus. Prove also that the equation \[ \frac{l}{r}(1+e\cos\alpha)^2 = \cos(\alpha-\theta) + e\cos(2\alpha-\theta) \] represents a circle which passes through the origin and touches the conic at the point where \(\theta=\alpha\).
By the methods of abridged notation or otherwise, prove that if two conics have each double contact with a third, their chords of contact with the third conic, and a pair of their chords of intersection with each other, will all meet in a point and will form a harmonic pencil.
In a triangle \(ABC\), \(D, E\) and \(F\) are the middle points of the sides \(BC, CA, AB\) respectively and \(Y, Z\) are the feet of the perpendiculars from \(B, C\) on to the opposite sides. \(YZ\) meets \(FD, DE\) respectively in \(M, N\). Prove that the circles through \(E, Y, N\) and \(F, Z, M\) have \(EF\) as a common tangent and \(AD\) as their radical axis.
Show that the inverse of a circle with regard to a point in its plane is a circle or a straight line. In Hart's linkage four rods \(PQ, QS, PR, RS\) are linked together at \(P, Q, R\) and \(S\) and lie in one plane so that \(Q, R\) are on the same side of the line \(PS\): the lengths are such that \(PQ=RS\) and \(PR=QS\). Show that, if the middle point of \(PQ\) be fixed, the middle points of \(PR\) and \(QS\) describe inverse figures with regard to it.
Prove that the mid points of the diagonals of a complete quadrilateral are collinear. Any line \(L\) meets the diagonals \(BD, AC\) of a parallelogram \(ABCD\) in \(X, Y\). \(X'\) is the point on \(BD\) harmonically conjugate to \(X\) with respect to \(B\) and \(D\), and \(Y'\) is the point on \(AC\) harmonically conjugate to \(Y\) with respect to \(A\) and \(C\). Prove by projection or otherwise that \(X'Y'\) and \(L\) are harmonically conjugate with respect to the pair of lines through their intersection parallel to the sides of the parallelogram.