If a variable line 3x + 4y - λ = 0 is such that the two circles x ^...

Given circles ${x}^{2} + {y}^{2} - 2x - 2y + 1 = 0 ⇒ {\left(x - 1\right)}^{2} + {\left(y - 1\right)}^{2} = {1}^{2}$ and

${x}^{2} + {y}^{2} - 18x - 2y + 78 = 0 ⇒ {\left(x - 9\right)}^{2} + {\left(y - 1\right)}^{2} = {2}^{2}$

Thus, centers of two circles are \left(1, 1\right) \text{ and } \left(9, 1\right)$and radii are$1 \text{ and } 2$respectively.</p><p>Since, center of circles are on the opposite side of line$3x + 4y - \lambda = 0$.</p><p>$⇒ \left(3 + 4 - \lambda\right)\left(27 + 4 - \lambda\right) < 0$</p><p>$⇒ \left(7 - \lambda\right)\left(31 - \lambda\right) < 0$</p><p>$⇒ \lambda \in \left(7, 31\right) \ldots \ldots . \left(i\right)$<br /><br />Since, circles should not intersect given line.</p><p>Hence, \left|\dfrac{7 - \lambda}{\sqrt{{3}^{2} + {4}^{2}}}\right| \ge 1 \text{\text{ and } } \left|\dfrac{31 - \lambda}{\sqrt{{3}^{2} + {4}^{2}}}\right| \ge 2$

$⇒ \left|7 - \lambda\right| \ge 5 \text{\text{ and } } \left|31 - \lambda\right| \ge 10$

$⇒ \lambda \le 2$ or $\lambda \ge 12 \ldots \ldots . \left(ii\right)$

and $\lambda \le 21$ or $\lambda \ge 41 \ldots \ldots .\left(iii\right)</p><p>Thus,$\lambda \in \left(i\right) \cap \left(ii\right) \cap \left(iii\right)

$⇒ \lambda \in \left[12,21\right]$