Question

Find the equation of the directrix of the parabola
9 x2 - 6xy + y2 - 13 x + y + 8 = 0.​

Answer 1

S≡9x  

2

−6xy+y  

2

+18x−6y+8=0

This equation is of the form :

S≡ax  

2

+2hxy+by  

2

+2gx+2fy+C=0 _____ (A)

Reverting the main equation

S≡9x  

2

+2(−3)xy+(1)y  

2

+2(9)x+2(−3)y+8=0

Compairing this equation with equation (A), we get

a=9;b=1;c=8;h=−3;g=9;f=−3

Distance between two parallel lines is given by :

d=2  

a(a+b)

g  

2

−ac

​  

 

​  

 or d=2  

b(a+b)

f  

2

−bc

​  

 

​  

 

So, d=2  

a(a+b)

9  

2

−ac

​  

 

​  

=2  

a(a+1)

(9)  

2

−9(8)

​  

 

​  

 

=2  

9×10

81−72

​  

 

​  

=2  

9×10

9

​  

 

​  

 

⇒d=  

10

​  

 

2

​