1887.] A. Mukhopadhyay— ilie7??oir on Tlane Analytic Geometry . 321 



Hence, remembering that the discriminant is a translation-invariant, we 

 can at once generalize the theorem to the case where the tangents are 

 drawn from any point, viz., the angle between the tangents is real, if 

 the discriminant is negative ; but we have shewn that, if the tangents 

 are real and the point outside the curve, the discriminant and the point- 

 function must have different signs, so that, as the discriminant is negative, 

 the point-f unction must be positive ; hence, finally, we have the very 

 simple 



Theorem. — Any point is outside a conic, on the curve, or inside it, 

 according as the point-function is positive, zero, or negative. 



§§. 22 — 23. Inclinations of tangents to conies. 



§. 22. Theorem. — We shall now prove a theorem which shews 

 how some well-known properties of the circle and the ellipse are corre- 

 lated. 



Consider the conic 



i4-^ _ (^^) 



where ^^ is essentially indeterminate in sign and value. The tangents 

 at any two points (^i, 2/i)> (^2» 2/2) ^^® 



and their chord of contact is 



a^ + f^ -"^"^ h^ ^^ ^^^) 



Hence, if 0, <p, ij/ be the angles of inclination of the two tangents and of 



their chord of contact to a directrix, we have 



2 



tan 0=-y.. ^^ (89) 



^^=-i^ (^^) 



Substituting for y^, y^ from (89) and (90) in 



we have 



^1 = -Tr== , H = —/ =^^^^ (02), (93) 



\/a^ + Z>2 tani^^ ^ifi + h^ iim^^ <p 



41 



