If 2 k x+3 y-1=0,2 x+y+5=0 are conjugate lines with respect to the circle x^2+y^2-2 x-4 y-4=0 , then k=

Step-by-step solution

Let point (x₁, y₁ ) lie on the line 2 k x+3 y-1=0 . x^2+y^2-2 x-4 y-4=0 Given circle, x^2+y^2-2 x-4 y-4=0 Equation of chord of contact from (x₁, y₁ ) to the circle aligned & x x₁+y y₁- 2 (x+x₁ ) 2 - 4 (y+y₁ ) 2 -4=0 & x (x₁-1 )+y (y₁-2 )-x₁-2 y₁-4=0 aligned Now, 2 x+y+5=0 and 2 k x+3 y+1=0 are conjugate line 2 x+y+5=0 and x (x₁-1 )+y (y₁-2 ) -x₁-2 y₁-4=0 are coincide. aligned & x₁-1 2 = y₁-2 1 = x₁+2 y₁+4 -5 = & x₁=2 +1, y₁= +2 & x₁+2 y₁+4=-5 aligned Put x₁, y₁ in x₁-2 y₁-4=-5 k aligned & 2 +1+2 +4+4=-5 =-1 & x₁=-1