A conic is the set of A conic is the set of points formed as the lines formed as the meets of corresponding joins of corresponding pairs of lines in two pairs of points in two projective pencils. projective pencils. Five points determine Five lines determine a conic. a conic. A conic can be A conic can be represented in point represented in line coordinates by a coordinates by a quadratic equation. quadratic equation. A point lies on a conic A line lies on a conic if and only if exactly if and only if exactly one of the lines on it on of the points on it lies on the conic. lies on the conic.
Moreover, the constructions and equations corresponding to these statements are dual In particular, We can construct a conic from five lines with the dual of the construction of a conic from five points. Similarly, We can discuss the line equations of a conic just as we discussed the point equations.