not as easy as abcd

My solution involves coordinate geometry.

the first statement can be re written as -

$\frac{c^2}{a^2 + b^2 } = 8$

so, the distance of the line $ax + by + c = 0$ from $(0,0)$ is $\sqrt{8}$ .

so $ax + by +c = 0$ is a tangent to the circle $x^2 + y^2 = 8$ .

the second statement can be written as -

$\frac{(a + b + c )^2}{a^2 + b^2} = d$

so $\sqrt{d}$ is the distance of the line from the point $(1,1)$ .

for $d$ to be minimum, the distance of line from $(1,1)$ should also be minimum. The minimum distance will be along the radius which passes through $(1,1)$ . The minimum distance would thus be the radius minus the distance of point $(1,1)$ from the origin.

$\sqrt{d} = \sqrt{8} - \sqrt{2} = \sqrt{2}$

so $d = \boxed{2}$ .