If a variable line, 3x + 4y -lambda = 0 | vedclass

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Question

$3x + 4y - \lambda  = 0$

$\left( {7 - \lambda } \right)\left( {31 - \lambda } \right) < 0$     (since centres lie opposite s

Vedclass · Accepted Answer

$[12, 21]$

Vedclass · Answer

$3x + 4y - \lambda  = 0$

$\left( {7 - \lambda } \right)\left( {31 - \lambda } \right) < 0$     (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 $\lambda  \ge 41\,\,\,\,\,\,\,\,\,\,\,.....\left( 3 \right)$

$\left( 1 \right) \cap \left( 2 \right) \cap \left( 3 \right)$

$\lambda  \in \left[ {12,21} \right]$

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

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