140 



ANALY 1 K' 



VII. 



Fio.04. 



SECANTS, TANGENTS, AND NORMALS 



8L Definitions of secants, tangents, and normals. A st might 

 liiu- will, in general, intersect any given curve in two or man 



distinct points ; it is then called a 

 secant line to the curve. L-t l\ 

 and P a be two successive points of 

 intersection of a secant lim- l\P 2 Q 

 with a given curve LP^P^ ... A - 

 if this secant line be rotated about 

 the point P 1 so that P 2 approaches 

 P l along the curve, the limiting 

 position P^T which the secant approaches, as P^ approaches 

 coincidence with P v is called a tangent to the curve at that 

 jtoint. This conception of the tangent leads to a method, <f 

 extensive application, for deriving its equation, the so- 

 called "secant method." * 



Since the points of intersection of a line and a curve are 

 found (Art. 39) by considering their equations as simulta- 

 neous, and solving for x and y, it follows that, if the line is 

 tangent to the curve, the abscissas of two points of intersec- 

 tion, as well as their ordinates, are equal. Therefore, if the 

 line is a tangent, the equation obtained by eliminating x or 

 y between the equation of the line and that of the curve 

 must have a pair of equal roots. 



If the given curve is of the second degree, then the equa- 

 tion resulting from this elimination is of the second degree, 

 and the test for equal roots is well known (Art. 1*) ; but if 

 the given equation is of a degree higher than tho second, 

 other methods must in general be used. 



A straight line drawn perpendicular to a tangent and 



For illustration, see Art. 84. 



