If a variable line,


If a variable line, $3 x+4 y-\lambda=0$ 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 $\lambda$ is the interval :-

  1. $[12,21]$

  2. $(2,17)$

  3. $(23,31)$

  4. $[13,23]$

Correct Option: 1


Centre of circles are opposite side of line



$\lambda \in(7,31)$

distance from $S_{1}$

$\left|\frac{3+4-\lambda}{5}\right| \geq 1 \Rightarrow \lambda \in(-\infty, 2] \cup[(12, \infty]$

distance from $\mathrm{S}_{2}$

$\left|\frac{27+4-\lambda}{5}\right| \geq 2 \Rightarrow \lambda \in(-\infty, 21] \cup[41, \infty)$

so $\lambda \in[12,21]$

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