If the lines 2 x+3 y+12=0, x-y+k=0 are conjugate with respect to the parabola y^2=8 x , then k is equal to

Step-by-step solution

Given, conjugate lines are 2 x+3 y+12=0 ...(i) and x-y+k=0 ...(ii) We know, that two lines are said to be conjugate with respect to a curve, if each passes through the pole of the polar of that curve. Let (x₁, y₁ ) be the pole of parabola y^2=8 x It's polar is y y₁=4 (x+x₁ ) 4 x-y₁ y+4 x₁=0 2 x- ( y₁ 2 ) y+2 x₁=0 ...(iii) On comparing Eqs. (i) and (iii), we get -y₁ 2 =3 y₁=-6 and 2 x₁=12 x₁=6 Pole (x₁, y₁ )=(6,-6) Eq. (ii) also passes through pole (6,-6) 6-(-6)+k=0 k=-12