88 ANALYi >V/.;/7,T [Cii. v. 



EXERCISES 



1. The perpendicular from the origin upon a certain line is 5; this 

 perpendicular makes an angle of * with the /-axis; what is the equation 

 of the line? 



2. If in equation [13] p is increased while a remains the same, what 

 is t he effect upon the line? If a be changed while/* remains th>- >amr, 

 what is the effect? 



3. A certain line is 3 units distant from the origin, and makes an angle 

 of 120 with the z-axis; what is its equation? 



4. Given a = 30, \\liat must be the length of p in order that the lin<> 

 //A' (see Fig. 44 a) shall pass through the point (7, 2) ? 



5. A line passes through the point (~3, ~4), and a perpendicular upon 

 it from the origin makes an angle of 225 with the z-axis. What is the 

 equation of this line? 



6. In Fig. 44 a draw through M a line parallel to II K, meeting 0.\ in 

 R\ then draw through P a perpendicular to MR, meeting it in Q; by 

 means of the figure so constructed derive equation [13] anew. 



56. Summary. The results of Arts. 51-55 may be briefly 

 summarized thus : 



The position of a straight line is determined by : (1) two 

 points through which it passes ; (2) one point and tin- <lin < - 

 tion in which the line passes through this point. Under (1) 

 there is the special case in which the two given points are 

 one on the ar-axis and the other on the y-axU. I'm In- (2) 

 there are two special cases : (a) when the given point is on 

 an axis (the y-axis say), and (#) when the point is given by 

 its distance and direction from the origin, while the line 

 whose equation is sought is perpendicular to tin- lim- which 

 connects the given point to the origin. 



Corresponding to these two general and three special cases, 

 there have been derived five standard forms of the equation 

 of the straight line, viz.: equations [9], [10], [U], [12], 

 and [13]. 



It may be remarked that equations [9] and [10] are inde- 

 pendent of the angle between the coordinate axes, while [11], 



