If a variable line, 3x+4y - λ = 0 is such that the two circles x2 + y2 - 2x - 2y + 1 = 0 and x2 + y2 - 18x - 2y + 78 = 0 are on its opposite sides, then the set of all values of λ is the interval :

Question

If a variable line, 3x+4y - λ = 0 is such that the two circles x2 + y2 - 2x - 2y + 1 = 0 and x2 + y2 - 18x - 2y + 78 = 0 are on its opposite sides, then the set of all values of λ is the interval : (A) [12, 21] (B) (2, 17) (C) (23, 31) (D) [13, 23]

Tardigrade Admin · Accepted Answer

Centre of circles are opposite side of line

(3 + 4 - λ) (27 + 4 - λ) < 0

(λ - 7) (λ - 31) < 0

λ \in (7, 31)

distance from

S_{1}

∣ ∣ \frac{3 + 4 - λ}{5} ∣ ∣ \geq \Rightarrow λ \in (- \infty, 2] \cup [(12, \infty]

distance from

S_{2}

∣ ∣ \frac{27 + 4 - λ}{5} ∣ ∣ \geq 2 \Rightarrow λ \in (- \infty, 21] \cup [41, \infty)

so

λ \in [12, 21]

Q. If a variable line, 3x+4y−λ=0 is such that the two circles x2+y2−2x−2y+1=0 and x2+y2−18x−2y+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]

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​ ∣∣​53+4−λ​∣∣​≥⇒λ∈(−∞,2]∪[(12,∞] distance from S2​ ∣∣​527+4−λ​∣∣​≥2⇒λ∈(−∞,21]∪[41,∞) so λ∈[12,21]

Q. If a variable line, $3 x + 4 y - λ = 0$ is such that the two circles $x^{2} + y^{2} - 2 x - 2 y + 1 = 0$ and $x^{2} + y^{2} - 18 x - 2 y + 78 = 0$ are on its opposite sides, then the set of all values of $λ$ is the interval :