If the equation 7 x^2-14 x y+p y^2-12 x+q y-4=0 represents a pair of parallel lines then the value of p^2+q^2-p q is

Step-by-step solution

Given equation of pair of lines is aligned & 7 x^2-14 x y+p y^2-12 x+q y-4=0 & a =7, ~b = p , c =-4, ~h =-7, ~g =-6, f = q 2 aligned The lines are parallel. aligned & h^2=a b & (-7)^2=7 p & p=7 aligned Now, abc +2 fgh - af ^2- bg ^2- ch ^2=0 aligned 7(7)(-4)+2 ( q 2 ) & (-6)(-7) & -7 ( q 2 )^2-7(-6)^2-(-4)(-7)^2=0 aligned aligned & -196+42 q- 7 q^2 4 -252+196=0 & q^2-24 q+144=0 & (q-12)^2=0 & q=12 & p^2+q^2-p q = 49+144-84 = 109 aligned