\(A\) is a fixed point outside a given fixed circle, and \(P\) is any point on the circumference. The line \(AF\) perpendicular to \(AP\) meets the tangent at \(P\) in \(F\). If the rectangle \(FAPQ\) is completed, prove that the locus of \(Q\) is a straight line.
Any irregular polygon is circumscribed about a circle. Prove that the perimeter of the polygon bears to the perimeter of the circle the same ratio as the area of the polygon to the area of the circle. Prove also that the same theorem is true for a polyhedron circumscribed about a sphere, if perimeter is replaced by surface, and area by volume.
Prove that the inverse of a system of non-intersecting coaxal circles with respect to a limiting point is a system of concentric circles with the inverse of the other limiting point as centre. \(U\) and \(V\) are two non-intersecting circles and \(A\) is a limiting point of the coaxal system of which they are members. A circle drawn through \(A\) and touching \(V\) at \(F\) meets \(U\) in \(P\) and \(Q\). Prove that \[ \frac{PF}{QF} = \frac{AP}{AQ}. \]
\(S\) 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.