Let curve f(x, y) is locus of point (x, y) which satisfy |2 x 2 y 6 α 2 β 3 λ -1 2 1|+2| x y 1 4 λ 2 α 3 β 2 -1 2|+|11 -2 λ 72 2α|-81 β =0 ∀ α, β, γ ∈ R. If minimum distance between curve f(x, y) and line 4 x-3 y+12=0 is d, then the value of d is

Question

Let curve f(x, y) is locus of point (x, y) which satisfy |2 x 2 y 6 α 2 β 3 λ -1 2 1|+2| x y 1 4 λ 2 α 3 β 2 -1 2|+|11 -2 λ 72 2α|-81 β =0 ∀ α, β, &gamma; ∈ R. If minimum distance between curve f(x, y) and line 4 x-3 y+12=0 is d, then the value of d is (A) 1 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

2 x(2 β-6 λ)-2 y(α+3 λ)+6(2 α+2 β)+2 x(4 α+3 β) -2 y(8 λ-6 β)+2(-4 λ-4 α)+22 α+144 λ-81 β=0 ∴ α(8 x-2 y+26)+β(10 x+12 y-69) +λ(-12 x-22 y+136)=0 . beginarraylx-y+13=0 ∴ 10 x+12 y-69=0 text and 6 x+11 y-68=0 endarray x=-(3/2), y=7 ∴ f(x, y) is point (-(3/2), 7) d=|(4((-3/2))-3(7)+12/5)|=3