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Question:

If a variable line, 3x+4y−λ=0 is such that the two circles x²+y²−6x−6y+1=0 and x²+y²−8x−6y+78=0 are on its opposite sides, then the set of all values of λ is the interval :- [12,21] (2,17) (23,31) [13,23]

(23,31)

[13,23]

[12,21]

(2,17)

Solution:

Centre of circles are opposite side of line
(3+4−λ)(27+4−λ)<0
(λ−7)(λ−31)<0
λ∈(7,31)
distance from S1 |3+4−λ|/5 ≥1 ⇒ λ∈(−∞,2]∪[12,∞)
distance from S2 |27+4−λ|/5 ≥2 ⇒ λ∈(−∞,21]∪[41,∞)
so λ∈[12,21]