The radius of a circle, having minimum area, which touches the curve `y=4-x^2` and the lines `y=|x|` is :

Options

A. `2 (sqrt(2) + 1)`
B. `2 (sqrt(2) - 1)`
C. `4 (sqrt(2) - 1)`
D. `4 (sqrt(2) + 1)`

Answer

C

Explanation

Let the radius of circle with least area be r.
Then, then corrdinates of centre = (0, 4 - r) .

since, circle touches the line y = x in first quadrant
`:. |(0 - (4 - r))/(sqrt(2))| = r implies r - 4 = +- r sqrt(2)`
`implies r = (4)/(sqrt(2) + 1)` or `(4)/(1 - sqrt(2))`
But `r =! (4)/(1 - sqrt(2))` `[ :' (4)/(1 - sqrt(2)) lt 0]`
`:. r = (4)/(sqrt(2) + 1) = 4 (sqrt(2) - 1)`