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

Answer: (A)

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