A circle `x^2+y^2+4x-2sqrt(2)y+c=0` is the director circle of the circle `S_1a n dS_1` is the director circle of circle `S_2,` and so on. If the sum of radii of all these circles is 2, then the value of `c` is `ksqrt(2)` , where the value of `k` is___________

Answer

4

Explanation

The radius of the given circle `x^(2)+y^(2)+4x-2sqrt(2)y+c=0` is `ssqrt(4+2-c) = sqrt(6-c)=a` (let).
Now, radius of circle `S_(1)=(a)/(sqrt(2))`
Radius of circle `S_(2)=(a)/(2)`
and so on .
Also, `a +(a)/(sqrt(2))+(a)/(2)+ .......+oo=2` (Given )
Then `a((1)/(1-1//sqrt(2)))=2`
or `(asqrt(2))/(sqrt(2)-1)=2`
or `a=2-sqrt(2)=sqrt(6-c)`
or `4+2-4sqrt(2)=6-c`
or `c=4 sqrt(2)`