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If a variable line, $3x + 4y -\lambda = 0$ is such that the two circles $x^2 + y^2 -2x -2y + 1 = 0$ and $x^2 + y^2 -18x -2y + 78 = 0$ are on its opposite sides, then the set of all values of $\lambda $ is the interval
$(2, 17)$
$[13, 23]$
$[12, 21]$
$(23, 31)$
Solution
$3x + 4y – \lambda = 0$
$\left( {7 – \lambda } \right)\left( {31 – \lambda } \right) < 0$ (since centres lie opposite side)
$\lambda \in \left( {7,13} \right)\,\,\,\,\,\,\,\,\,\,\,…….\left( 1 \right)$
$\left| {\frac{{7 – \lambda }}{5}} \right| \ge 1$ and $\left| {\frac{{31 – \lambda }}{5}} \right| \ge 2$
$\left| {7 – \lambda } \right| \ge 5\,$ and $\,\,\left| {31 – \lambda } \right| \ge 10\,$
$\lambda \le 2\,\,$ or $\lambda \ge 12\,\,\,\,\,\,\,\,\,……\left( 2 \right)$
and $\lambda \le 21\,\,$ or $\lambda \ge 41\,\,\,\,\,\,\,\,\,\,\,…..\left( 3 \right)$
$\left( 1 \right) \cap \left( 2 \right) \cap \left( 3 \right)$
$\lambda \in \left[ {12,21} \right]$
