TANGENT AND NOEMAL. EXAMPLES. 287 



265. If the equation to a curve be given in the form 

 F(x,y)-c = Q, 



the equation to the tangent at the point (x, y} } will be 

 /JTf /JTT 



V-rtf+^-jf-o ............... CD; 



and the equation to the normal 



, , v dF . , . dF , . 



If we consider x, y , as constant, equation (1) combined 

 with F(x,y} = c, will give the co-ordinates of the points 

 where the tangents drawn from the point (x, y'} meet the 

 curve represented by F (x, y) = c ; and equation (2) combined 

 with F (x, y) = c will give the co-ordinates of the points 

 where the normals drawn from the point (x, y'} meet the 

 curve represented by F (x, y} = c. 



Since the equations (1) and (2) are independent of c, they 

 will represent the geometrical loci of the points where the 

 curves which we obtain by ascribing different values to c in 

 the equation F(x, y} = c, are met by their tangents or their 

 normals respectively, which pass through the point (x, y'}. 

 Thus, if we want to draw tangents from the point (x, y") to 

 any one of the curves F (x, y] =c, we must construct the 

 curve 



and determine where it intersects the particular curve 

 F(x, y) = c which we are considering ; join the point or 

 points of intersection with the point (of, y'} and we have 

 the required tangent or tangents. Similarly, we may draw 

 normals from (x r , y'} to any one of the curves F (x, y} = c. 



EXAMPLES. 



1. In the curve y (x - 1) (x - 2) = x - 3, shew that the tan- 

 gent is parallel to the axis of x at the points for which 

 x = 3 V2. 



