1 1 I ANALYTIC OEOME1 /; > ... V. 



i 



' 7. // \, G ;'. Jiiul /'= 'J; 



A U _9-8+-|?-Z=0; 



:! tip- first member can be factored. 



The factors may be found as follows: transposing, dividing l.y 2, ami 

 completing the square of the ar-terms, the equation may be written in 



the form *' + "-^x + (^f-*)* = fjj(y - 2y + 1); 



therefore the given equation, divided by 2, may be written in the form, 



i... (* + y + i)(*- !y+ i) =0; 



hence the locus of the original equation consists of the straight lines 



x + y + 1 = and 2ar - 3y + 7 = 0, 

 which agrees with the result of Art. 66. 



EXERCISES 



Prove that the following equations represent pairs of straight lines ; 

 find in each case the equations of the two lines, the coordinates of their 

 point of intersection, and the angle between them. 



1. 6y a -^-ar-h30y + 36 = 0. 



2. ** - 2xy - 3y* + 2x - 2y + 1 = 0. 



3. x a - 2 xy sec a + y 2 = 0. 



4. ;r + 6jry + 9y + 4*+ 12y-5 = 0. 



5. For what value of k will the equation 



X 2 _ 3 xy + y + ID* - 10 y + k = 

 represent two straight lines? 



SUGGESTION: Place the discriminant (A) equal to zero, and thus find 

 k = 20. 



Find the values of k for which the following equations represent pairs 

 of straight lines. Find also the equation of each line, the point of inter- 

 section of each pair of lines, and the angle between them. 



6. 6z + 2*xy + 12y* + 22z + 31y + 20 = 0. 



7. 12z* + 36;r + ** + 6;c + 6 + 3=0. 



