y=logex is inverse of y=ex so it is mirror image of each other with respect to y=x line. Slope of tangent to y=ex curve ‌
dy
dx
=ex Slope of tangent to y=logex curve, ‌
dy
dx
=‌
1
x
Both tangents are parallel to y=x line for minimum distance condition. ∴ Slope of y=x line = Slope of both the tangent. ∴‌
dy
dx
=ex=1⇒ex=e0=x=0 ∴y=ex=e0=1 and ‌
dy
dx
=‌
1
x
=1⇒x=1 ∴y=loge1=0 ∴ tangent at (0,1) point of y=ex curve and tangent at (1,0) point of y=logex curve are parallel. ∴ Minimum distance between point (0,1) and (1,0) is =√12+12=√2 ∴a=√2 So, b2=‌