If a variable line, 3x + 4y -- [ ] = 0 is such that the two circles x2 + y2 -- 2x -- 2y + …

Question

If a variable line, 3x + 4y -- $$\lambda $$ = 0 is such that the two circles x2  + y2 -- 2x -- 2y + 1 = 0 and x2 + y2  -- 18x -- 2y + 78 = 0 are on its opposite sides, then the set of all values of $$\lambda $$ is the interval :

Accepted Answer

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 S1

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

distance from S2

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

so  $$\lambda  \in \left[ {12,21} \right]$$

Solutions · 1