3. Given the axes of a conic in position, and also a tangent with its point of contact, to construct the foci and determine the length of the axes.
4. Given the tangent at the vertex of a parabola, and two other tangents, to find the focus.
5. The locus of the center of a circle touching two given circles is a conic with the centers of the given circles for its foci.
6. Given the axis of a parabola and a tangent, with its point of contact, to find the focus.
7. The locus of the center of a circle which touches a given line and a given circle consists of two parabolas.
8. Let F and F' be the foci of an ellipse, and P any point on it. Produce PF to G, making PG equal to PF'. Find the locus of G.
9. If the points G of a circle be folded over upon a point F, the creases will all be tangent to a conic. If F is within the circle, the conic will be an ellipse; if F is without the circle, the conic will be a hyperbola.
10. If the points G in the last example be taken on a straight line, the locus is a parabola.
11. Find the foci and the length of the principal axis of the conics in problems 9 and 10.
12. In problem 10 a correspondence is set up between straight lines and parabolas. As there is a fourfold infinity of parabolas in the plane, and only a twofold infinity of straight lines, there must be some restriction on the parabolas obtained by this method. Find and explain this restriction.