If a variable line 3x+4y-=0 is such that the two circles x^2+y^2-2x-2y+1=0 and x^2+y^2-18x…

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:

Accepted Answer

Centres $(1,1)$ (radius $1$) and $(9,1)$ (radius $2$).

Opposite sides: $(3+4-\lambda)(27+4-\lambda)<0\Rightarrow(\lambda-7)(\lambda-31)<0\Rightarrow\lambda\in(7,31)$.

Line must not intersect either circle: $\dfrac{|7-\lambda|}5\ge1\Rightarrow\lambda\le2	ext{ or }\lambda\ge12$; $\dfrac{|31-\lambda|}5\ge2\Rightarrow\lambda\le21	ext{ or }\lambda\ge41$.

Intersecting all conditions: $\lambda\in[12,21]$.

Solutions · 1