As we stated in Chapter 4, a fundamental idea in studying a conic Q is to understand how it intersects lines. It is however not just the intersections of Q with a single line which are significant for its geometry, but its intersections with pencils of lines. That is a major theme of this text, which we introduced in Chapter 7 by studying the intersections of Q with parallel pencils of lines. In this chapter we develop the theme by studying how Q meets a general pencil of lines through a point on Q itself. That leads to a central geometric idea, the ‘tangent’ to Q at a point, representing the best possible first-order approximation. In Section 9.3 we introduce the companion idea of the ‘normal’ to Q at a point, the line through that point perpendicular to the tangent. In the next three chapters we will use the material developed so far to look at the three main conic classes of ellipses, parabolas, and hyperbolas in more detail. Each has distinctive features, which are best discussed within the context of their class.

Tangent Lines

Consider the pencil of lines through a fixed point W on a conic Q. Think of another point W′ on Q, and consider the line L through W, W′. (Figure 9.1.) The idea is that as W′ moves along Q into coincidence with W, so L will tend towards a limiting position, the ‘tangent’ line at W.