Prove that the locus of a point, the lengths of the tangents from which to two fixed circles are in a constant ratio, is a circle. Prove also that the locus of a point, at which two fixed segments of the same straight line subtend equal angles, is a circle.
Prove that the common chords of an ellipse and a circle are in pairs equally inclined in opposite senses to the axes. Two circles are drawn through \(P\) and \(Q\), points on an ellipse, to touch the ellipse and \(R, R'\) are the respective points of contact: prove that the angle at which the circles cut is twice the angle which \(RR'\) subtends at \(P\).
Two figures in a plane are directly similar but not similarly situated and the points \(A, B\) in one figure correspond respectively to \(A', B'\) in the other: the lines \(AB\) and \(A'B'\) meet in \(C\): prove that the second point (\(O\)) of intersection of the two circles \(CAA'\) and \(CBB'\) is the point about which the one figure may be rotated so as to be placed similarly situated to the other with \(O\) as the centre of similarity.
Determine the locus of the centre of a circle which touches two given coplanar circles. Three given spheres are external to one another and a variable sphere touches all three externally: prove that the centre of the sphere lies on one branch of a hyperbola.
Prove that, if \(S\) be a fixed point and \(L\) a fixed line in a plane and the line \(PS\) meet \(L\) in the point \(R\), a projective transformation is set up between the points \(P\) and \(p\) by taking the point \(p\) on the line \(PS\) produced so that \(RP \cdot Rp = RS^2\). Show that this transformation converts any conic, focus \(S\) and directrix \(L\), into the circle on its latus rectum as diameter. Show also that the tangent to the conic at \(P\) meets the tangent to the circle at \(p\) on a line parallel to \(L\) and twice the distance of \(L\) from \(S\).
Prove that one parabola has double contact with each of two circles and that its focus is midway between the two centres of similitude of the circles.
Prove that, if two tangents to the ellipse \(x^2/a^2 + y^2/b^2 = 1\) intersect in the point \((X, Y)\), the point of the intersection of the corresponding normals has coordinates \[ \left( \frac{(a^2-b^2)X(b^2X^2-a^2Y^2)}{b^2X^2+a^2Y^2}, \frac{(a^2-b^2)Y(a^2X^2-b^2Y^2)}{b^2X^2+a^2Y^2} \right). \] (Note: This transcription is based on the visual appearance of the formula, which seems more plausible than the OCR'd version.) Deduce that the line joining the intersection of two tangents to the intersection of the corresponding normals is parallel to the major axis, if \[ Y=0, \quad \text{or} \quad (a^2-2b^2)X^2 - a^2Y^2 = a^2(a^2-b^2). \]
Find the tangential equation of the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Show that a parabola can be drawn to touch the axes of an ellipse, any two normals of the ellipse and the chord joining the feet of the two normals.
Prove that any line is in general a tangent to one of a given family of confocal conics and a normal to one of the family. Prove that the points of contact of tangents from the point \((f,g)\) to the confocals of the family \(x^2/(a^2+\lambda) + y^2/(b^2+\lambda) = 1\) lie on the cubic curve \[ (fy-gx)(x^2+y^2-fx-gy) = (a^2-b^2)(x-f)(y-g) \] and that the six points of intersection of this cubic with any one of the conics are the points of contact of the two tangents and the feet of the four normals from the point \((f,g)\).
Prove that the locus of the centre of a variable conic passing through four fixed points in a plane is a conic and that the locus cannot be a parabola unless one of the four fixed points is at infinity.