A line drawn from (0,h) will have the equation

y-h=mx

For this line to be tangent to given circle, distance of line from origin should be equal to the radius of the circle,i.e 4

Applying distance formula of a point to line,

|h|/(1+m^2)^0.5 = 4

Now, we have another condition, area of triangle APB should be minimum.

Lets find out points A and B first, these are the points of intersection of line with x axis, so x= -h/m, and by symmetry, other point is +h/m

A=Area of triangle = 1/2 *base* height= 1/2 * 2h/m * h = h^2/m and this expression needs to be minimised.

From the relation between m and h we had obtained using distance formula,

h^2= 16(1+m^2)

Putting this in A, A= 16(1+m^2)/m= 16 (1/m + m) >= 16(2)

where minimum value occurs at m=1

so, h^2= 16(2)=32, h=32^0.5