JEE PYQ: Circle Question 43
Question 43 - 2019 (12 Jan Shift 2)
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:
(1) $(2, 17)$
(2) $[13, 23]$
(3) $[12, 21]$
(4) $(23, 31)$
Show Answer
Answer: (3)
Solution
$C_1(1,1), r_1 = 1$; $C_2(9,1), r_2 = 2$. On opposite sides: $(3+4-\lambda)(27+4-\lambda) < 0 \Rightarrow (7-\lambda)(31-\lambda) < 0 \Rightarrow \lambda \in (7, 31)$. Distance $\ge r$: $\frac{|7-\lambda|}{5} \ge 1 \Rightarrow \lambda \ge 12$ or $\lambda \le 2$. $\frac{|31-\lambda|}{5} \ge 2 \Rightarrow \lambda \le 21$ or $\lambda \ge 41$. Intersection: $[12, 21]$.
BETA
