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.
A is a point on a given circle. Shew how, with ruler and compasses, to find another point P on the circle such that the sum of the distances of P from A and from the tangent at A shall be equal to a given length not greater than twice the diameter of the circle.
A circle is drawn to cut the auxiliary circle of an ellipse at right angles and to touch the ellipse at P. The normal PG to the ellipse at P cuts the circle again in Q. Prove that GQ is equal to the radius of curvature of the ellipse at P.
Prove that if P be any point of a hyperbola whose foci are S and H, and if the tangent at P meets an asymptote in T, the angle between that asymptote and HP is twice the angle STP.