Question

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 :-

• A. $[12,21]$
• B. $(2,17)$
• C. $(23,31)$
• D. $[13,23]$

Answer 1

The correct option is A $[12,21]$

Centre of circles are opposite side of line

$(3+4-\lambda )(27+4-\lambda )<0$
$(\lambda -7)(\lambda -31)<0$

$\lambda \in (7,31)$
distance from $S_1$

$\left|\frac{3+4-\lambda }{5}\right|\ge 1\Rightarrow \lambda \in (-\infty ,2]\cup [12,\infty )$
distance from $S_2$

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

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