Official solution
Let the radius of circle with least area be r.
Then, the coordinate of the center = (0, b)
\[ \therefore \] The equation of circle be
x2 + (y -- b)2 = r2
Distance of perpendiculur from (0, 4) to y = x line = r
\[ \Rightarrow \] \[\left| {{{ - b} \over {\sqrt 2 }}} \right| = r\]
\[ \Rightarrow \] b = \[{\sqrt 2 r}\]
Circle passes through (0, 4),
\[ \therefore \] 0 + (4 -- b)2 = r2
\[ \Rightarrow \] 4 - b = r
\[ \Rightarrow \] 4 - \[{\sqrt 2 r}\] = r
\[ \Rightarrow \] r = \[{4 \over {\sqrt 2 + 1}}\] = \[4\left( {\sqrt 2 - 1} \right)\]