The radius of a circle, having minimum area, which touches the curve y=4−x² and the lines y=|x| is.

Let the centre be (0,k)
Now, we know the radius as r=4−k
Now the radius will be equal to the perpendicular distance from the lines.
4−k=√k²/2
=>k=4√2/(1+√2)
Radius, r=4−k=4−4√2/(1+√2)
r=4(1−√2/(1+√2))
r=4(1+√2−√2)/(1+√2)
r=4/(1+√2)
r=4(√2−1)
Therefore, r=4(√2−1)