The range of parameter `' a '` for which the variable line `y=2x+a` lies between the circles `x^2+y^2-2x-2y+1=0` and `x^2+y^2-16 x-2y+61=0` without intersecting or touching either circle is `a in (2sqrt(5)-15 ,0)` `a in (-oo,2sqrt(5)-15 ,)` `a in (0,-sqrt(5)-10)` (d) `a in (-sqrt(5)-1,oo)`

Explanation

The given circles are
`C_(1) : (x-1)^(1)+(y-1)^(2)=1,r_(1)=1`
and `C_(2) : (x-8)^(2)+(y-1)^(2)=4,r_(2)=2`
The line `y-2x-a=0` will lie between these circles if the centers of the circles lie on opposite sides of the line, i.e.,
`(1-2-a)(1-16-a)lt0`
or `a in (-15,-1)`
The line will not touch or intersect the circles if
`(|1-2-a|)/(sqrt(5))gt1,(|1-16-1|)/(sqrt(5))gt2`
or `|1+a| gt sqrt(5),|15+a| gt 2sqrt(5)`
`i.e., `agt sqrt(5)-1` or `alt sqrt(5) -1`
and `agt 2 sqrt(5)-15` or `alt - 2 sqrt(5)-15`
Hence, the common values of a are `(2 sqrt(5)-15,-sqrt(5)-1)`.