2. A point moves along a straight line. Show that its polar lines with respect to two given conics generate a point-row of the second order. 3. Given five points, draw the polar of a point with respect to the conic passing through them, without drawing the conic itself. 4. Given five lines, draw the polar of a point with respect to the conic tangent to them, without drawing the conic itself. 5. Dualize problems 3 and 4. 6. Given four points on the conic, and the tangent at one of them, draw the polar of a given point without drawing the conic. Dualize. 7. A point moves on a conic. Show that its polar line with respect to another conic describes a pencil of rays of the second order. _Suggestion._ Replace the given conic by a pair of protective pencils. 8. Show that the poles of the tangents of one conic with respect to another lie on a conic. 9. The polar of a point _A_ with respect to one conic is _a_, and the pole of _a_ with respect to another conic is _A’_. Show that as _A_ travels along a line, _A’_ also travels along another line. In general, if _A_ describes a curve of degree _n_, show that _A’_ describes another curve of the same degree _n_. (The degree of a curve is the greatest number of points that it may have in common with any line in the plane.) CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS *107. Diameters. Center.* After what has been said in the last chapter one would naturally expect to get at the metrical properties of the conic sections by the introduction of the infinite elements in the plane. Entering into the theory of poles and polars with these elements, we have the following definitions: The polar line of an infinitely distant point is called a _diameter_, and the pole of the infinitely distant line is called the _center_, of the conic. *108.* From the harmonic properties of poles and polars, _The center bisects all chords through it (§ 39)._ _Every diameter passes through the center._ _All chords through the same point at infinity (that is, each of a set of parallel chords) are bisected by the diameter which is the polar of that infinitely distant point._ *109. Conjugate diameters.* We have already defined conjugate lines as lines which pass each through the pole of the other (§ 100). _Any diameter bisects all chords parallel to its conjugate._ _The tangents at the extremities of any diameter are parallel, and parallel to the conjugate diameter._ _Diameters parallel to the sides of a circumscribed parallelogram are conjugate._ All these theorems are easy exercises for the student. *110. Classification of conics.* Conics are classified according to their relation to the infinitely distant line. If a conic has two points in common with the line at infinity, it is called a _hyperbola_; if it has no point in common with the infinitely distant line, it is called an _ellipse_; if it is tangent to the line at infinity, it is called a _parabola_.