The number of integral values of $\lambda$ for which the variable line $3 x+4 y-\lambda=0 \quad$ lie

Question

The number of integral values of $\lambda$ for which the variable line $3 x+4 y-\lambda=0 \quad$ lies between the circles $x^2 +y^2-2 x-2 y+1=0 \quad$ and $x^2+y^2-18 x-2 y+78=0$ without having any common points with any circle is equal to 2 k , then value of k is--

Professor PC Thomas Classes · Answer

$$ \begin{aligned} & S_1: x^2+y^2-2 x-2 y+1=0 \quad C_1(1,1) \ & r_1=1 \ & S_2: x^2+y^2-18 x-2 y+78=0 \quad C_2(9,1) \ & r_2=2 \ & C_1 M_1 \geq r_1 \ & \frac{|3+4-\lambda|}{5} \geq 1 \ & |7-\lambda| \geq 5 \ & 7-\lambda \geq 5 	ext { or } \lambda-7 \geq 5 \ & \lambda \leq 2 	ext { (reject) or } \lambda \geq 12 \ & 	ext { similarly } C_2 M_2 \geq r_2 \ & \frac{|27+4-\lambda|}{\sqrt{3^2+4^2}} \geq 2 \ & 31-\lambda \geq 10, \lambda \leq 21 \end{aligned} $$ Hence $12 \leq \lambda \leq 21$ It is given that there should be no common point with circles, so we will not include 12 and 21. Possible values of $\lambda$ are $\{\mathbf{1 3}, \mathbf{1 4}, \mathbf{1 5}, \ldots ., \mathbf{2 0}\}$ $$ \begin{aligned} & 2 k=8 \ & k=4 \end{aligned} $$