A \ !/>//< BMQM1 i I;Y [CM. 



reduced. These standard forms are tin- simplest for stu-lv- 

 ing the curves ; but the student must discriminate carefully 

 between general results ami those which huhl onlv when the 

 equation is in the standard form. 



IV. TAV.KNTS, NOKM.M>. POLARS, DIAMETERS, BTO, 

 121. Since the equation 



always represents a conic whose axes are parallel to the 

 coordinate axes, and since by giving suitable values to tin- 

 constants A) B, (?, JP, and (7, equation (1) may represent any 

 such conic, therefore, if the equations of tangents, normals, 

 polars, etc., to the locus of equation (1) can be found, inde- 

 pendent of the values that A, B, etc., may have, these equa- 

 tions will represent the tangents, etc., when any special 

 values whatever are given to the constants involved. 

 In the next few articles such equations will be derived. 



122. Tangent to the conic 



Ax* + By* + 2Gx + 2Fy + C = O 



in terms of the coordinates of the point of contact : the secant 

 method. The definition of a tangent has already Keen ^i\c 

 t. 81), and the method to be employed her. in finding 

 4 nation is the one which was used in Art. * \. That 

 article should now be carefully re-read. 



Let the given conic, i.e., the locus of the equation, 



2ax + 2Fy+ (7=0, . . . (1) 



be represented by the curve BHK\ and let PI = (a^ yO be 

 the point of tangency. 



