Two concentric circles of which smallest is `x^(2)+y^(2)=4`, have the difference in radii as d, if line y=x+a cuts the circles in real points, then d lies in the interval

Options

A. `(-oo, -2-(1)/(sqrt(2)))uu(-2+(1)/(sqrt(2)),oo)`
B. `(-2+(1)/(sqrt(2)),2+(1)/(sqrt(2)))`
C. `(-oo, 1-(1)/(sqrt(2)))uu(1+(1)/(sqrt(2)),oo)`
D. `(1-(1)/(sqrt(2)), 1+(1)/(sqrt(2)))`

Answer

A

Explanation

It is given that the difference in the radii of two concentric circles is d. So, let the equations be
`x^(2)+y6(2)=4 " " ...(i)`
and, `x^(2)+y6(2)=(2+d)^(2) " " ...(ii)`
If the line y=x+1 cuts the circles in real points, then `x^(2)+(x+1)^(2)=(2+d)^(2)` must have real roots.
`:. 4-8+8(2+d)^(2) gt 0`
`rArr 2(2+d)^(2)-1 gt 0`
`rArr (2+d)^(2)-((1)/(sqrt(2)))^(2) gt 0`
`rArr (2+d-(1)/(sqrt(2)))(2+d+(1)/(sqrt(2))) gt 0`
`rArr d lt - 2-(1)/(sqrt(2)) or, d gt - 2 + (1)/(sqrt(2))`
`rArr d in (-oo, -2 - (1)/(sqrt(2))) uu (-2 + (1)/(sqrt(2)), oo )`