Given circles x2+y2−2x−2y+1=0⇒(x−1)2+(y−1)2=12 and
x2+y2−18x−2y+78=0⇒(x−9)2+(y−1)2=22
Thus, centers of two circles are (1,1)&(9,1) and radii are 1&2 respectively.
Since, center of circles are on the opposite side of line 3x+4y−λ=0.
⇒(3+4−λ)(27+4−λ)<0
⇒(7−λ)(31−λ)<0
⇒λ∈(7,31)…….(i)
Since, circles should not intersect given line.
Hence, |\frac{7-\lambda }{\sqrt{{3}^{2}+{4}^{2}}}|\geq 1 & |\frac{31-\lambda }{\sqrt{{3}^{2}+{4}^{2}}}|\geq 2
\Rightarrow |7-\lambda |\geq 5 & |31-\lambda |\geq 10
⇒λ≤2 or λ≥12…….(ii)
and λ≤21 or λ≥41…….(iii)
Thus, λ∈(i)∩(ii)∩(iii)
⇒λ∈[12,21]